Lecture Notes in
Mathematics
Edited by A. Dold and B. Eckmann
626
NumberTheoryDay
Proceedings of the Conference
Held at Rockefeller University,
New York 1976
Edited by M. B. Nathanson
Springer-Verlag
Berlin Heidelberg NewYork 1977
Editor
Melvyn B. Nathanson
Department of Mathematics
Southern Illinois University
Carbondale, IL 62901/USA
AM S Subject Classification s (1970): 10 D 15,10 E 20,10 L 05,10 L 10,12 A 70
ISBN 3-540-08529-7 Springer-Verlag Berlin Heidelberg New York
ISBN 0-387-08529-7 Springer-Verlag New York Heidelberg Berlin
This work is subject to copyright. All rights are reserved, whether the whole
or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying
machine or similar means, and storage in data banks.
Under § 54 of the German Copyright Law where copies are made for other
than private use, a fee is payable to the publisher, the amount of the fee to be
determined by agreement with the publisher.
© by Springer-Verlag Berlin Heidelberg 1977
Printed in Germany
Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr.
2140/3140-543210
On 4 M a r c h 1976 t h e l ~ o c k e f e l l e r U n i v e r s i t y h o s t e d a o n e - d a y c o n f e r e n c e on
number theory.
The lectures
were as follows:
S. C h o w l a , " L - s e r i e s
and elliptic
curves;" P. ErdSs, "Combinatorial p r o b l e m s in n u m b e r theory;" P. X. Gallagher,
" P r i m e s and zeros in small intervals;" C. J. M o r e n o , "Explicit formulas in the
theory of automorphic forms;" M .
B. Nathanson, "Oscillations of additive bases;"
and A. Selberg, "l~emarks on multiplicative functions. " T h e field of n u m b e r theory
w a s thus fairly broadly represented.
T h e papers in the present v o l u m e are accounts,
several in expanded versions, of m o s t of these lectures.
M.
B. Nathanson, w h o w a s
the original instigator of this n u m b e r theory day, has kindly offered to serve as
editor.
W e take this opportunity to m a k e
record of our gratitude to our distinguished
speakers for their participation.
M. S c h r e i b e r
1 S e p t e m b e r 1977
TABLE
OF CONTENTS
i. S. Chowla,
L-series and elliptic curves
I
Z.
P. ErdBs,
P r o b l e m s and results on combinatorial n u m b e r
3.
C. J. M o r e n o ,
4.
M.
5.
A. Selberg, R e m a r k s
theory III
Explicit formulas in the theory of automorphic f o r m s
B. Nathanson,
Oscillations of bases in n u m b e r
on multiplicative functions
theory and combinatorics
43
73
217
232
oC-series and elliptic c u r v e s
S. C h o w l a
Introduction
A. Selherg and S. C h o w l a p r o v e d in Crelle I s Journal, 1965, that if there
w e r e a tenth i m a g i n a r y quadratic field Q(V/T-~,
corresponding
with c l a s s - n u m b e r
i,
then the
~-series
o~s)
= 2~ ( d ) n - s[ ( _d 1 i s t h e K r o n e c h e r
n
n
symbol]
1
would be negative at
Subsequently,
problem
1
s = "~ ,
contradicting
and independently,
A.
( a l r e a d y s t a t e d by G a u s s ) .
quadratic
the "extended Riemann hypothesis".
B a k e r a n d H. M. S t a r k s o l v e d a n o u t s t a n d i n g
Namely,
fields with class-number
1.
there are exactly
9 imaginary
G a u s s u s e d t h e l a n g u a g e of b i n a r y q u a d r a t i c
f o r m s instead of that of quadratic fields.
The problem
(still unsolved) of the existence (or, otherwise
of rational
points on a given elliptic curve
y
Z
w a s c o n s i d e r e d by Euler, Poincar~,
Swinnerton-Dyer,
in m o s t
of certain H e c k e
~-series
3
=x
+ax+b
Mordell,
remarkable
Nagell and m a n y
conjectures,
others.
Birch and
related the o r d e r of the zero
(formed with"GrBssencharaktere")
associated with
certain elliptic c u r v e s to the " M o r d e l l - W e i l "
again, w e see the i m p o r t a n c e of the point
IIecke
o~- series
Let
~K(S)
rank of these curves. So, h e r e
1
s = -~ for the values of Dirichlet and
~(s).
denote the D e d e k i n d zeta function
(N~) -s
72r
where
z,Z runs over all integral ideals of the algebraic n u m b e r
field
K.
It is a n a l m o s t u n s p o k e n conjecture - I believe implicit in the w o r k
Selberg a n d C h o w l a m e n t i o n e d a b o v e - that
of
if K
is of degree
Z
(there is a m p l e n u m e r i c a l
Serre (in a letter to the author) m a d e
evidence by l~osser, L o w ,
Purdy).
the surprising discovery that
r~K(~) = o
for a certain field
Q(~'-,
~/~-).
K
of degree
S =~
This field is a quadratic extension of
A proof, with a different example,
All these e x a m p l e s
1
8.
in the study of the
w a s published by A r m i t a g e .
s h o w the special interest that attaches to the point
~-series
of Dirichlet and Hecke.
A n o t h e r a p p r o a c h to the conjectured non-vanishing
of Dirichlet
~-series
o0
>2 X (n)
s
(s> 0;× #×0 )
1 n
on the real line
s > 0 is provided by a paper (also in Crelle v s Journal, v o l u m e
dedicated to H. Hasse) by the author,
M.
J. de L e o n and P. Hartung.
R e c e n t notes by John Friedlander and the author (Acta Arithmetica,
Vol.
Z8, P a r t 4, 1976 and G l a s g o w Math. Journal, Vol. 17, 1976) again s h o w the import1
s = ~ for ~-~-series, (in the next line, X denotes a character
ance of the point
(rood k))
o0
~-~s) = >2 x(n)s [X ~XO]
1
Let
d
be a p r i m e
number
h(d)
of the f o r m
of Q(~r'~), for
x
Z
n
+ i. S. C h o w l a
x > Z6, is > i.
conjectured that the class-
In fact, for x = Z6,
h(677) = I
This is analogous to the G a u s s conjecture that h(-d) > 1 for all d > 163
is square-free).
Finally,
recent unpublished w o r k
zeta function of cubic fields,
when
K
~K(S)
studied in her P h . D .
seems
is a cubic field. H e r w o r k
of Epsteinrs zeta function
of M .
(where
C o w l e s on the D e d e k i n d
to indicate that
combines
the results of special cases of
thesis (Penn State University 1976) with the estimates
l
Z(s) at s = ~ m a d e by Selberg and C h o w l a in Crellets
d
Journal, 1965, cited above.
T h e rest of this p a p e r is divided into five parts:
P a r t I. S o m e
remarks
on the coefficients
oo
x
IT,, (l-x n)Z(l-xHn)2
L
c
of the parity of c
n
This w o r k is joint with M .
for the first time.
P a r t Z.
1
the p r o b l e m
in the expansion of
0o
:Ec
i
l
Here,
n
xn
n
is solved, as far as w e k n o w ,
Cowles.
The congruence
e
is studied.
Here
0-(n) =
n
~ d.
~ ~(n)(mod 5)
[ (n, ii) = i]
This w o r k is joint with J. Cowles.
din
P a r t 3.
l~temarks on D e d e k i n d s u m s .
A n e w expression is obtained for the c l a s s - n u m b e r
of i m a g i n a r y quadratic
fields.
This is closely linked to recent w o r k of Hirzebruch.
P a r t 4.
On
Fermatls
last t h e o r e m .
This is a n account of recent w o r k with P. C h o w l a ,
linking the study of
F e r m a t ' s equation
x p + yP : z p
with the p r o b l e m
(p_>5)
of "rational" points on
y Z : xp + ~ 1
P a r t 5.
l~ecent unpublished w o r k with D. Goldfeld on relations b e t w e e n Epstein's
zeta functions a n d D e d e k i n d zeta functions.
1.
O n the coefficients
c
in the expansion
n
co
x
co
I[
(1-x n)z(1-x lln)z =2
1
§ O.
Let
p(n)
c xn
n
1
be defined (Euler) by
o0
1
p(n)x n co
1
The problem
following
result
II ( t - x n)
l
of the parity
concerning
of
p(n)
the value
of
is still unsolved,
c
( r o o d Z).
Let
but we will prove
p
be a prime,
the
then
n
a)
b)
Z divides
c
if
P
p ~ 1,3,4,5,
If
p ~ Z, 6 , 7 , 8 , 1 0
( m o d 11)
Z
9 ( r o o d 11),
then
Z divides
c
iff
p = u
+ llv Z
(u,v~Z).
P
An announcement
Ample
support
in the classical
the Legendre
Corollary:
of this
paper
symbol.
result
has
recently
for this result
of Shimura
appeared
is provided
[ 3] .
[ 1] .
in Trotter's
In the following
We give two proofs
table for
(~),
of the foilowing
(p<Z000)
P
an odd prime,
is
q
c
coroliary.
T h e polynomial
4(x3-x Z) + I
ha s
i)
exactly
ii)
exactly
iii)
one linear
3
factor
Two
n
p ~ n ( r o o d 11) w h e n
(~)
provided
no linear
factors
( r o o d p)
power
series in x
if
p --- r ( r o o d li)
integers
u
co
x
and
'',, ( I - x n)
1
From
CO
Z =Z
c xn
I
n
where
r
(~)
r
(~-)
= +l,
= -1 .
= +1,
provided
v.
are said to be congruent
of the difference is congruent to 0 (rood Z).
follows
if
linear factors
( m o d p) i f p ~ r ( m o d ll) w h e n
Z
p = u + llv Z for some integers
u and v.
3p = u Z + llv Z for some
§ I.
( m o d p)
( m o d Z) if every coefficient
oo
23
cO
c x
n
=
x
I I (l-xZn)(l-xZZn)(m°d
n
1
Z)
.
(i)
1
Now
3uZ_u
cO
oo
l[ (l-xn) = I + 23 (-l)U[ x
1
1
So changing
x to x
Z
+x
3uZ+u
Z
]
(2)
Z
I
1
oo
3uZ+u]
(l-xZn) = 1 + ~B (-l)U[ x 3uZ-u + x
1
(3)
T h e right side of (3) can be written
oo
Ux3U2
(-i)
+u
-00
Thus
II
~
i
Changing
x to x
ii
(l-xZn) =23c° (-l)Ux3UZ+U--- ~S°° x3UZ+U
-oo
-oo
(4) by
o0
(i) and
3uZ+u+ll(3vZ+v)
X
(rood Z) .
(6)
(6)
(6u+l)Z+ll(6v+l)Z
oO
~ c x
n
1
Now
(rood Z) ,
(5) produces
I I (l-xZn)(l-xZZn) - 23
1
u, vc Z
From
(4)
in (4), w e obtain
oo
co ll(3vZ+v)
[ I (I-xZZn) =- 23 X
1
-cO
and multiplying
(rood Z) .
(7) m e a n s
n
23
U~ V E Z
the following:
x
iZ
(rood 2) .
(7)
Theorem
h
n
where
is the n u m b e r
d
(8)
(rood Z)
=-d
c
n
of solutions of
n
(6u+l) 2+ii (6v+l) Z
n
§ g.
From
Theorem
Theorem
=
(9)
IZ
1 w e shall deduce.
Z:
c
( m o d Z)
~0
n
if
n 2--Z, 6,7,8,10
Proof:
T h e result will follow f r o m
(rood Ii)
Theorem
1 once it is s h o w n that w h e n
(I0)
( r o o d ll)
n - Z,6,7,8,10
we have
d
= 0
(il)
.
n
Now
by
(9), (ii) follows i m m e d i a t e l y f r o m
(I0), since if d
w e r e not
0
n
(iz)
IZn = (6u+l) Z + ll(6v+l) Z
would h a v e solutions.
Since
IZ ---1 (rood ll) a n d
Ii ---"0 (rood ii), (IZ) w o u l d
i m p l y that
n E (6u+l) Z
contradicting
§ 3.
(I0).
N o t e that in the result of § Z,
Theorem
(iz)
(rood ll)
n w a s not restricted to be a p r i m e ,
Z is stronger than the result stated in the introduction.
n will be a p r i m e
p.
thus
In this section
r
Theorem
3:
If p---r (rood ii) w h e r e
( ~ ) = +I,
then
Z
divides
c
if
P
Z
p
for s o m e
integers
We
u
and
= u
+ llv z
v.
need a number
of e l e m e n t a r y
lemmas
from
the theory of binary quadratic
fo r m s .
Lemrna
h
If p ~'1,3,4,5,9
Z
+ llv Z
(rood 11) then either
i) p = u
or
ii) p = 3 u Z + Z u v + 4 v Z
Z
but not both.
Exaraple:
[ Thus
either
p
or
p = 47 = ii(i)Z + (6) Z.
choice of u
and
v
3p
is of the f o r m
We
cannot h a v e
for this w o u l d m e a n
igp
=
u
+ llv Z.]
p = 3u Z + Zuv + 4v Z for a n y
that
564
=
(6u+_Zv) 2 + 4 4 v 2 .
It is easily verified that this is not the case.
Recalling T h e o r e m
1 for
n = p,
with
p =- 1 , 3 , 4 , 5 , 9
(rood II) as in Section
Z we have
c
---d
P
where
is the n u m b e r
d
(rood Z)
P
of solutions of equation (9) for
n = p,
i.e. Of
P
(18)
iZp = (6u+l) Z + ll(6v+l) Z .
From
the v e r y i m p o r t a n t Satz
sentations of a n u m b e r
forms
n
Z04
of L a n d a u
by the w h o l e
of a given discriminant, w e
[Z] , giving the n u m b e r
set of primitive
r e d u c e d binary quadratic
see that (18) does in fact h a v e
p=u
of repre-
Z
solutions if
Z + fly Z
a n d only one solution if
p = 3u Z + Zuv + 4 v Z .
Hence
not divide
Cp,
Z
if p = u Z + llv Z a n d the opposite case, i.e., Z
P
holds iff p = 3u Z + Zuv + 4vZ; w h e n
p ='='I,3,4,5,9 (rood II).
divides
e
does
4.
It is a special (proved) case of the Weil conjectures that the n u m b e r
of
N
P
solutions of the c o n g r u e n c e
y
is, for
p } ii,
Z
= 4(x3-x z) + 1 (rood p)
given by
N
= p
P
Clearly
(i) N
-
c
.
(19)
P
is e v e n if 4(x3-x Z) + 1 is irreducible
( m o d p),
while
ii) N
P
is odd if 4(x 3
from
(19),
c
Z) + 1 has exactly one or three linear factors
(rood p).
Thus,
is odd in case i) and even in case ii). This leads to the proof of
P
the corollary stated in § 0.
Cowles'
proof of the corollary c o m e s
of the D e d e k i n d zeta function of the field K
arising f r o m
Q
f r o m a study
by adjoining a root
of 4(x3-x Z) + 1 = 0.
§6.
In conclusion w e recall a f e w other properties of the coefficients
i) p + 1 - c
- 0
c :
n
(rood 5)
P
ii) c
mn
= c
m
c
n
Relations of i) with s o m e
if (re,n) = i.
identities of R a m a n u j a n
are pointed out in a recent
(unpublished) joint note of J. C o w l e s and S. Chowla.
l~eferences
i. S. C h o w l a and M.
x ~
I
2.
Cowles, O n the coefficients
c
n
in the expansion
(l_xn)Z(l_xll n)Z = ~ c x n , J. reine angew. Math. Z9Z(1977), 115-116.
1
n
E. Landau, " E l e m e n t a r y N u m b e r
Theory," Trans. by J. E. G o o d m a n ,
Chelsea,
N e w York, N . Y . , 1966.
3.
G. Shimura, A reciprocity law in non-solvable extensions, J. l~eine Angew.
Math. Z21(1966), Z09-ZZ0.
10
Z.
Remarks
on c o n g r u e n c e
c
n
c~
2
n=l
of the coefficients
~
c xn = x
n
c
m
c
n
(l _ x n )Z( l _ x l l n )g
n
n=l
It is a striking property of the
These
properties
in the expansion
-- c
c
mn
defined a b o v e that they satisfy
n
if (re, n) = 1 .
,
coefficients a r e also i m p o r t a n t in the study of reciprocity lairs in non-
solvable extensions
of the rationals,
e.g.,
see S h i m u r a ' s
p a p e r [Z] .
Another
striking p r o p e r t y is given by
(A)
c
---p + 1 (rood 5),
for p r i m e
p ~ ii .
P
This is implicit in the theory of elliptic curves,
with the W e i l conject*~res.
See the p a p e r [3]
There
(A)
is a m p l e
of S h i m u r a ' s
support of
(B)
was
extend
c
0- (n)=
E d a.
a
dl n
in the table for
p < Z000
formulas
unsuccessful.
n
(A)
by a s i m p l e a r g u m e n t
---~ (n) (rood 5),
1
When
a = I,
for
to
(n, ll) = 1
w e will write
o(n)
in place of o l(n).
But w e h a v e results of interest w h i c h
recursive m e t h o d ,
late the coefficients
consequence
by A.
proof of
of l ~ a m a n u j a n w o u l d aid us to do this.
point of v i e w m i g h t lead to an e l e m e n t a r y
a n obvious
Mathematicae.
by Trotter at the end
our original p u r p o s e to give a direct but e l e m e n t a r y
that s o m e
§1.
by Tare in fnventionaes
p a p e r [Z] .
In this p a p e r w e
where
recently studied in connection
Cn.
(A).
m e n t i o n h e r e the estimate
w o r t h recording.
thought
Our
O u r p a p e r also includes
to calcu-
[ cp I < Z V'-~ w h i c h is a
hypothesis for plane c u r v e s over finite fields, p r o v e d
Weil.
A recursive f o r m u l a for the
Proceed
We
h a v e so far b e e n
apparently not r e c o r d e d in the literature,
We
of the R / e m a n n
proof of
seem
We
(A).
It
by "logarithmic
c .
n
differentiation"
of the equation
11
oo
E
oo
n
n=l
log
(Ecnxn)
= log
x+
n=l
ZE log(1-x
Enc
n
n) + ZE log(1-xlln),
x
n-1
1
-
Ec
n (l - x n )z(1 - x lln )z
c xn = x
n
n
M u l t i p l y the last relation b y
x
x
x
nx
-
2
E
-
n=l
-
l-x
n
and multiply by
lln- I
nx
ZZE - 1 - x lln
-
Ec
x
n
n
n
Enc
Then,
n
x n = (Ecnxn
)(1-ZE
nx
lln
- 22E
l-x
n
nx
----~n
l-x
)"
since
co
n
co
E - n -x
= E o- ( n ) x n
n=l
1-x n
n=l
and
co
E
n=l
~nc
x
n
= (~c
n
lln
co
E 0(1%)x lln,
-
nx
lln
l-x
xlln
x )(l-ZZ~(n)xn-zz~(n)
I% n
n=l
).
Thus
comparing
coefficients of
x
n
o n b o t h sides:
nc
n
= c
n
- Z
n-1
E
t=l
c 0-(n-t) - ZZ
t
c o-(v) .
Z
u
u+llv=n
u, v > l
The
desired
formula
is
nc
n
= c
n
- Z
E
c
u+v=n
n
~(v)
- ZZ
u, v>l
This formula
f
and
g
to m e a n
l e a d s u s to i n t r o d u c e
Y
u+v=n
u, v>l
f(u)g(v).
w i t h itself will b e t a k e n up.
E
u+l]v=n
c
u
~(v).
u, v_>l
the t e r m
"convolution"
of t w o functions
In S e c t i o n 3, the c o n v o l u t i o n of the f u n c t i o n
0-
12
§ 2.
T h e proof of (B)
From
(A)).
Shimura's paper [ Z] ,
0o
(C)
(assuming
e
X
e
ms = (1-11-s)
m=l
m
.II (1- ~ + -'~gs )-1
p~ll
p
p
Z
= (l+ll-S+ll-ZS+ll-3S+...)
where
x
= c p
P
P
-s
- p
[-Zs
From
this,
3
II ( l + x + x + x + . . . )
P P P
p~ll
for
p # 11,
n
c
=
p
Zn
c
are inctuded
-
gn+l
r
P
reZn-Zr
p
n
=
p
Both cases
>2 ( 1 ) r ( z n - r )
r=0
23 (-l)r(Zn+l-r)prcZn+l-Zr
r
p
r=0
in
rl
[7]
(D)
c n=
p
( 1 r n-r
rcn-Zr
- ) ( r )p
P
E
r=0
.
n
The key step in the proof
of
c
n
P
-= E p r ( r o o d 5),
r=0
for prime
p ~ 11,
is
the foliowing
n
E (_l)r(n;r)pr(p+l)n-Zr
r=0
Lemma.
Proof:
integers
Proceed
k < n.
by induction
Consider
on
=
n-
the ease
E pr
r=0
assume
when
the lemma
n = Zj,
holds for all non-negative
the case when
n
is odd is
simila r.
E
r=0
J
E
r=0
(_l)r(n;r)pr(p+l)n-Zr
j-1
( - 1 ) r ( Z J r r ) p r ( p + l ) ZJ- Z r = (- I)0 (~j)p0 (p+l) Zj +
E
r=l
.
.
(-l)r(Z3;r)pr(p+l) Z3-zr
73
(-l)J(ZJ.-J)pJ(p+l)Zj-Zj = (_l)O(Zj-l)pO(p+l) 2j
3
j-I
+ Z (-l)r[(ZJ-r-l)+(ZJrr?l)]pr(p+l)ZJ-Zr + (_l)J(Zj-~-l)pj (p+l) gj-zj
+
r=l
j-1
.
.
j
.
= N (_l)r(ZJ-rr-1)pr(p+l)ZJ-gr + >2 (_l)r(gj~.r/1)pr(p+i)ZJ-Zr
r=0
r=l
j-i
(_ 1)r((ZJ-1)- r)pr(p+l) (2J -1)- Zr
7"
= (p+i)
r
r=0
n-i
j-1
[--2-1
+ (-p) 7" (-1)t((ZJ[Z)-t)pt(p+t) (Zj-Z)-Zt = (p+l)
N (-1)r((n-1)-r)pr(p+l) ( n - 1 ) - z r
t=0
L
r=0
r
n-2
[--2-]
+ (-p)
n-i
(_l)t((n- Zt)- t)pt(p+l )(n-g)-Zt
Z
= (p+l)
~
t=0
n- z
p
r
- p
r=0
From
(A) and
(D),
it follows that c
~
n
pt
t=0
=
Z
pr
r=0
_=~(pn) ( m o d 5),
for p r i m e
p ~ Ii.
n
F r o m the c a s e
n
m =p ,
j u s t p r o v e d , it is i m m e d i a t e that
c
since the
c's
n
- ~(n)(mod 5),
for (n, ll) = 1 ,
are multiplicative and so is ~.
n
Finally note that f r o m f o r m u l a
(C),
it follows that c
not difficult to see that
= i. T h u s it is
lln
c
n
_ ~( n )(rood 5)
ii~
and
(=+l)c
- ~ ( n ) ( m o d 5)
n
where
ll~In but ii~+I ~ n.
From
the first of the two c o n g r u e n c e s above, together with the recursion
formula in the previous section, w e obtain the
Theorem.
~.
5"
~(-~--)(r(v) +
7
~(
u+v=n
u
u+llv=n
ii
u, v>l
u, v__>l
o~.+1
w h e r e Ii ill but ii z
~ i.
u
ii
u
)~(v) - Z(n-l)¢(--2---) ( m o d 5) ,
n
II
14
§ 3.
The
c o n v o l u t i o n of
(r w i t h its elf.
In this section w e
RamanujanVs
evaluate the c o n v o l u t i o n of
Collected Papers
where
1
o-(0) = ~ ~(-1).
~(u)cY(v) = i.(4)
zeta-function
= (z=)s~(l-s)
s = Z)
-2~(2)
1
~(-1) = - -
above,
1
~(o)
-
-
g4"
2;
~
= - Zw g
4~ 2
from
From
cr3(n) + ~(0)n~(n)
N o w t h e f u n c t i o n a l e q u a t i o n of t h e R i e m a n n
Sir
Hence
5).
~Z(z)
" ~(4)
Zr(s)~(s)cos 7
gives (setting
(mod
([I], p. 139), w e h a v e
rZ(z)
2]
u+v=n
u, v > 0
0- w i t h itself
6
Thus the formula
~(u)~(v) =
-
Z
1
lg
"
of l ~ a m a n u j a n g i v e s
z~(0)~(n) + ~1 (4 )Z -90
'~-~3(n)
i
- ~ n~(n)
.
u+v=n
u, v > l
*(u)*(v)
Thus
u+v=n
=
1
*(n) + ~15
- ~3(n) - ~n~(n).
57-
Hence,
t a k i n g t h i s e q u a t i o n (rood 5),
u, v>l
Z
o-(u)o-(v) ~ 3o-(n) - 3no-(n) (rood 5) ;
u+v=n
u, v>l
2
u+v=n
u, v > l
e ( u ) ~ ( v ) ~ Z(n-1)~(n) (rood 5) .
15
I~eferences
1.
S. l ~ a m a n u j a n , " C o l l e c t e d
2.
G. S h i m u r a ,
P.
V. S. A i y a r ,
Math.
3.
Papers
of S r i n i v a s a
a n d B. M. W i l s o n ,
A reciprocity
IL~manujan".
Chelsea,
law in non-solvable
New York,
extensions,
I~d. b y G. H. H a r d y ,
New York,
221(1966), 209-220.
J. Tare, T h e arithmetic of elliptic curves, Inv. Math.
196Z.
J. I~eine A n g e w .
23(1974), 179-206.
16
3.
§ I.
Let
h(d)
On Dedekind
d e n o t e the c l a s s - n u m b e r
sums
of the q u a d r a t i c field
Q(~/d)
and write
k
E ~k ~ (~k)
=t
s(h,k) =
{ where
I
%b(x) = x - [x] - -~ if x
~b(x) = 0
for the D e d e k i n d
Further
s u c h that
if x
is a n i n t e g e r }
sum.
let
t,u
x Z - d y Z = I,
b e the s m a l l e s t positive integral v a l u e s of x , y
Here
d
is a positive n o n - s q u a r e
I
bz_
/ d = b 0 - bll_
where
the
bls
the " u p p e r "
a r e integers
> Z
and
c o n t i n u e d fraction for
s
simple
....
integer.
1
bs_l -
respectively
Write
1
bs
is the length of the (smaLlest) p e r i o d in
~/d. A l s o let
v/d = a0 + i
al+
b e the o r d i n a r y
is not a n integer,
1
az+
c o n t i n u e d fraction for
1
" ' " +--at
d
w i t h p e r i o d length
t.
Write
B = bl + b z + --. + b s
A
= a t - at_ 1 + - ... + a I .
We s h a l l s k e t c h the p r o o f s of T h e o r e m s
l a n d 2, f r o m w h i c h T h e o r e m 3
is a n i m m e -
diate consequence.
Theorem
i. If d
is a p r i m e
-3(4)
then
3 - g-it + igs(t,u) = 3s - B .
u
]Example:
d = 7.
R.S. of (I) = 3. Z -
Here
t = 8, u = 3;
1
1
~/7 = 3 - ~-- ~-
(3+6) = - 3
L.S. of ( 1 ) - - 3 - 5 - - + - 5 -
{ *( ) + Z~(
)} = - 3 .
(l)
17
Theorem
Z.
If d
is a p r i m e
-3(4),
then
(z)
3s - B = - A
(many
when
d
examples,
is a p r i m e
this a n d c o m b i n i n g
Theorem
3.
including
~ 3(4)
and
theorems
If d
d = 1019, w e r e
h(d) = i,
c h e c k e d by P.
Hirzebruch
Chowla)
p r o v e d that
3h(-d) = A.
Using
1 a n d Z w e obtain
is a p r i m e
~'3(4)
then if h(d) = l
3 - Z_jt + IZs(t,u) = -3h(-d)
(3)
u
Examples:
This applies to all p r i m e s
O n e can check,
§ Z.
We
as a n e x a m p l e ,
d ---3(4)
the case
Dedekind
"Analytic N u m b e r
Theory"
1955), p. Z56.
Let
H-function.
1 a n d Z.
except
d = 79.
T h e y a r e b a s e d on the theory
Recall the following f r o m
(Tara Institute of F u n d a m e n t a l
a,b,c,d
I00,
d = 19.
shall sketch the proofs of T h e o r e m s
of the w e l l - k n o w n
less than
be positive integers with
l~ademacher's
Research,
ad - bc = i.
Bombay,
1954-
T h e n with
lm(~-) > 0
,aT+d,
log ~tc---~--~j = log ~(T)
1
c~+d
(4)
wi
+~log F-- +~(a+d)
-
wi s(d, c)
(there will be no d a n g e r of confusing the
d
h e r e with o u r previous
d).
O n the
other h a n d s u p p o s e
aT+b
cT+d - b0
Then
I
b~
i
b Z-
I
"'" - (bs+T)
one easily a r g u e s that
(aT+b)
1
log ~3 ~
- log 13(7) - ~ log(cr+d)
wi
--
wi
4 s +~-
(bo+bl+..°+bs)
(5)
18
§3.
We apply
so that
(4)
and
(5)
with (here
we write
d = N)
a =t,
b=Nu
c=u,
d=t
a d - b c = 1.
Let (wi~integral
b ' s ~ Z)
Nu
T
l
= b0 - b 1-
1
1
b Z-
bs
(bo=b s) •
T h e n (the bracketed portion is the "period")
1
1
~/N = b 0 - bl -
Comparing
obtain
§ 4.
(4)
and
(5)
(1) of T h e o r e m
Theorem
It states
ba -
in our special
1
...
case
b
(N
1
Zb
s-l-
s
is a prime
of t h e f o r m
4k+3)
we
1.
g is proved
that (see pages
by using the famous
Reciprocity
Z59 a n d Z57 of 1 K a d e m a c h e r J s
Law for Dedekind
Sums.
book cited above)
1 1 (_d c i /
s(d,c/+ s(c,d/ =- ~+TZ- c + ~ + c d
when
c,d > 0
and
(c,d)
law allows us to calcuiate
Also
s(-c,d)
can be built.
= -s(e,d).
= 1.
Since
s(c,d)
rapidly the values
These
are
has a period
of
s(c,d)
the main ideas
c
in
when
c
d,
and
the reciprocity
d
are
large.
o n w h i c h a p r o o f of T h e o r e m
3
19
Z
4.
§ I.
Write
(p
The
non-trivial
is a n o d d p r i m e
rational points in y
= 4x p + 1
> 3)
x p + yP = l
(F)
and
y
We
and
(H).
on
are
o n the c u r v e with
x = i,
on
(H)
main
y = 0
a non-trivial " Q - p o i n t " ]
converse
§ Z.
We first
Theorem
i.
.
and
x = O,
now
x = 0,
on a "Q-point"
on
(F)
but p e r h a p s
we
mean
or "trivial" Q - p o i n t s
y = I.
y = -1.
on
(F)
or
implies one on
(H)
(H)
will refer to
and conversely.
prove
F => H.
on
F
implies
one on
H.
From
= 1
on squaring
(xP-yP) x + 4(xy) p = 1
Set
x p - yP
= u
x y = -V.
So
"obvious"
Q)
(F)
not entirely trivial.
x p + yP
follows,
The
y = 1 and
A Q-point
is e l e m e n t a r y ,
T h a t is a Q - p o i n t
Proof.
y ~ Q.
(with coefficients in
in
are
result is that a [ f r o m
The
f(x,y) = 0
x c Q,
x = 0,
Our
(H)
of p o s s i b l e non-trivial " Q - p o i n t s "
B y a Q - p o i n t o n the " c u r v e "
(x, y)
Those
= 4x p + 1
a r e interested in the p r o b l e m
a point
(F)
Z
(i) b e c o m e s
(1)
20
u
So if x,y ~ Q
in (i), then
u,v { Q
2
(z)
= 4v p + 1 .
in ( 2 ) .
q.e.d.
So, the first half of our assertion at the end of 1 is trivial.
We n o w prove:
~3.
Theorem
Z.
H~>
F.
T o this end, w e set in
(Z),
ce
u
-f
2'
(3)
v=8
with
~,p,,~, 8 { z;
So
( ~ , P ) = ('l, 8) = 1
(4)
(Z) b e c o m e s
Z
Z = 4(~6 )P + 1
(5)
P
or
~26P
Since the r.h.s,
= 4~Zy p + ~ Z 6 P .
of (i) is - 0 ( ~ z) w e obtain
~Zl~26P.
(6)
But
(ce,~) = 1 and so,
~218P.
O n the other hand f r o m
(6), 6P(~2-~ Z) = 4~Z'f p.
8P]4p z
§ 4.
Case
We
n o w distinguish
A:
6,
in
So, since
(~/,6) = 1
(8)
Z cases:
(6), is odd
In this case it follows f r o m
(8) tha t
6Pip z
From
(7)
(7) and
(9)
{9} w e obtain
2 = 8p .
(10)
21
So
Z
=~ ,
with
5 = ~i
(n)
Pl ~ Z.
By (ill
(6)
becomes
= 4y p + p
(iZ)
Or
(13)
(~-~P)(~+~5 p) = 4y p .
Now
odd.
~
From
is odd from
(1) a n d
(10)
since
(1Z) ~1 i s o d d .
8
is odd.
Since
~
and
So from
~1 a r e
(11) a n d
odd,
(13),
it follows
a
is
from
(13) that
- ~1p = z'~p'
From
~ + ~1p = z~p I ~ i Y z ' z , ,q,t z = 81
(i4)
(14)
Z~ p = Z(,~Z-yi
P P)
(i5)
I. e.
P
P
P
~31 + Y1 = YZ
i.e.
(16)
(F) has the Q-point
~I
x=
1
Z'
Y-
Y
Thus, in case
Z
Y
(A)
(H) --~ F .
§5.
Case
We
(B): 8,
in
(6),
is e v e n .
shall use the notation
q~llM
to m e a n
that, with
q a prime,
we have
22
qlM
i.e. q
of
is the highest
In this case,
power
namely
of
q
B,
c+llM
but
dividing
M.
In our application,
q=Z.
let
zelrs
(17}
i. e.
5 = zcs1
Then, f r o m
(18)
(5 1 o d d ) .
(6)
413Zy p _- 5P{of2_[32).
(19)
Then
is odd
5'
Since
in
(19) h a s
Zp
[since
as a factor
and
(zo)
(~, 6) = 1] .
p > Z it follows
from
(19)
or
(ZO)
that
[3 i s e v e n
[so Of is odd, since
(of,[3) = i]
(18) w e n o w h a v e
From
zcP-elf[32
I.e.
(zl)
(21)
s erring
c=Zd.
(z2)
Z d p - l ] [ [3 .
(z3)
We have
So,
set
[3 = 2dP-l[31
5 = 22dst
the latter from
So
(18)
and
([31 odd),
(8 1 odd),
(ZZ).
(19) gives (cancelling out 22dp
f r o m both sides)
(24)
(25)
23
=
z
pl)
(26)
i. e.
Z6p
Z
1 =~ Yp+
Since
r.s.
zZdp-Z Z p
~161
(Z7)
of (Z7) is ~ 0(~iZ) w e get since ~ is p r i m e to ~,
Z
and so to ~i" that
p
(28)
~iTbl •
Also, f r o m
From
From
(Z6) since
(Z8) and
(26) and
6 is prime to ~/ (and so
6 1 is prime to "~) that
2
(29)
p
Z
61 = ~I"
(30)
(Z9),
(30),
u p = (=z-zZdP-Z~Z).
(31)
So
(~+ Z dp- i~i )(~ - Z dp- I~i) = U p .
Since
6 was even (hypothesis of case B, first line of §5), "~ is odd [since
I] so each factor on the ~. s. of (3Z) is odd using
are relatively p r i m e since
Thus, f r o m
(=,~)
=
(3Z)
('~,6 ) =
(21) above; also the Z factors
i.
(3Z)
+ zZdp-I~I = YIP
(331)
+ zZdp-I~I = -~P.
(33 Z)
[ %ve used, in (3Z), that if the product of Z relatively p r i m e nos. is a pth-power,
then each no. is a p-th power] .
Subtracting
(33Z) f r o m
(331) w e get
zZdP~I
(34)
24
But ~i
is a p-th p o w e r
from
(30),
~l: ~pl
From
(34)
and
say
[~l~ z]
(35);
P
P = (zZdNI)P .
Y2 - Yl
i° e.
(36)
(F) h a s the " Q - s o l u t i o n "
YI
YZ
X
-
Zk I
Thus
(35)
H
=> F
(above proof).
Since
Y -
F => H
F<=>
Corollary:
Fermatls
Last Theorem
(non-trivial)
(§Z)
it follows that
H.
is true if a n d only if the c u r v e
y
h a s no rational points
ZX I
Z
= 4x p + l
on it.
(37)
25
5.
O n the twisting of Epstein zeta-functions into
Hecke-Artin
§ I.
L - s e r i e s of K u m m e r
fields
T h e Epstein zeta-function
Z(s, C) = ~'
(ax 2+ bxy+ cy 2) s
w h e r e the s u m m a t i o n
is o v e r all integers
x,y
excluding
x = y = 0,
with associated
binary quadratic f o r m
Z
C = ax
Z
+ bxy + cy
a n d dis criminant
Z~ = b Z - 4ac < 0
is so defined for
l~e(s) > i,
and by analytic continuation over the w h o l e s-plane.
H e r e , by a b u s e of language,
Z
3
+ bxy + cy .
C
refers to the set of integers representable by
ax
We
integer).
a r e c o n c e r n e d with
Z(s, C)
in the special case w h e n
A
= -3k Z
(k,
an
In this case, G a u s s and D e d e k i n d noted the connections b e t w e e n these
functions a n d the law of cubic reciprocity.
More
explicitly, D e d e k i n d p r o v e d that
(Crelle's Journal, 1900)
Z~K(s)
~(s)
where
forms
=
i
~I
.
_
1
~I
(xZ+ZTyZ)S
(l)
(4xZ+ Zxy+TyZ)S
3
~i4(s) is the D e d e k i n d zeta-function for the field K = Q(~/2). Note that both
2Z
Z
2
x + Z7y
and 4 x + 2xy + 7y
h a v e discriminant -108 = -3.63 . This rela-
tion implies that for p r i m e s
p ---1 ( n o d 3),
Z is a cubic residue
( m o d p) if and
only if
p = x
This r e m a r k a b l e
result is due to Gauss.
Z
Z
+ Z7y .
We
quote f r o m D e d e k i n d ' s p a p e r (pp. Z06-
207 of his Collected Papers).
O b s ervatio venustis s i m a inductione facta
Z
es__~tl~esiduurn vel n o n R e s i d u u m
formae
3n + I,
prout
p
cubicum numeri primi
representablis est p e r f o r m a n
p
26
x x + Z7yy
vel
4 x x + Zxy + 7yy.
3
per
est R e s i d u u m
xx + Z43yy
vel n o n R e s i d u u m ,
au___~t4 x x + Zxy + 61yy
prout
p
representabilis es__~t
ve__! 7xx + 6xy + 36yy aut
9xx + 6xy + Z8yy.
(Note that the f o r m
§ Z.
Let
S
13x Z + 4 x y + 19y Z of discriminant
-972
h a s b e e n omitted~ )
be a set of integers such that
S = Cl[.J CZ~_) C 5 ... ~J C H
where
each
C. is also a set of integers and
J
C 1.... , C H
f o r m a multiplicative g r o u p
We
G.
define
co
D(s, C.) = E
J
n:l
w h e r e the
c.(n)
,I
ns
cj(n) are arbitrary c o m p l e x n u m b e r s ,
the Dirichlet series associated to
co
C..
3
Let
a
H
and
cj(n) = 0
if n ~ C.,3 to be
n = Z~ D(s, C . ) .
n=l n s
j =i
J
B y a "twist" of the left side of (Z), w e m e a n
(Z)
the n e w series
H
x (C.)D(s, C.)
j=l
~
J
a
where
X
is any non-trivial character of G.
In general, if >2.--Sn has a n E u l e r
n S
product, then the "twisted" series also has a n E u l e r product.
A s a n e x a m p l e of a twist let S
of the following
3 forms:
be the set of integers representable by a n y
27
C I = x Z + Z7y Z
C Z = 4 x Z + Zxy + 7y 2
C 3 = 4 x Z - Zxy + 7y Z
s o that
s = ciU czU ¢3"
(3)
T h e associated Dirichlet series
In this e x a m p l e ,
(E'
D(s,C.) are nothing but ]Epstein zeta functions.
J
the analogue of (Z) is
i
)(I-3-s+3. 9 -s) = Z ( s , C I) + Z(s, C Z) + Z(s, C3).
(xZ+3yZ) s
(4)
This is p r o v e d as follows:
In the s u m
i
I~ = E'
(5)
(xZ+3yZ) s
the variables
into 4
x,y
ranging over all integers excluding
classes, so that
R=A+B+C+D
where
A =
I
~'
31x
(xZ+3yz)s
Sly
B=
1~'
3Ix
i
(xZ+3yZ) s
3 ,y
31y
= ~(3-s_9 -s) ,
-R"
9
- s
x = y = 0,
can be divided
28
C :
i
~'
3~'x (xZ+3yZ)s
?[y
:
~' -
31y
2'
31~
3[y
= Z ( s , Cl) - 9-s1%,
D =
[
2;'
3~'y (xZ+3y2) s
3~y
=Z
---1(3)
/x
>2v +y--2(3)
E' ]
\~---I(3) y =-z(3)/
=2
~y(3)
Now,
3 Ix
3ty
in the s u m
i
x=-y(3 ) (xZ+3yZ) s
w e just m a k e
the transformation
X = U +
ZV t
y
:U-
V
w h i c h leads to
D = g(Z(s, Cz)-A).
Hence
(4) is proved,
by using the relation
1% : A + B + C + D.
Let
x (c l) = t x ~
I+ VZ~X (C z) =
x ( C 3) :
J
be a character of the composition group
quadratic f o r m s
of discriminant
-108,
Z
{CI, Cz, C 3 ]
of reduced primitive binary
satisfying the following relations
29
Z
C1
= CI
Z
C Z
: C3
C3 Z = C 2
CzC 3 : C 1 •
T h e structure of the a b o v e g r o u p is d e t e r m i n e d by the fact that if n, is in C. and
1
1
in C. then n.n. is in C.C..
3
J
ij
1 3
In Section 8 of this p a p e r it will be p r o v e d by a novel m e t h o d that the "twist"
n
of
(~,
1
) (i_3-s+3. 9-s)
(xZ+3yZ) s
is nothing but
%K(S)
This, of course,
gives D e d e k i n d ' s result
(i), and the m e t h o d
extends to m a n y
other cases.
§ 3.
W e w o u l d like to ask the follo~ving question.
E u l e r product) on
~K(S)
where
K = Q(%/~,
zeta functions of sub-fields of K u m m e r
Is any "twist" (which has an
a ratio of products of Dedekind's
A positive a n s w e r
fields?
is supported
by the e x a m p l e s that follow.
A Kurn~er
where
k
answer,
§4.
and
a
field is
are positive integers.
If the a b o v e question does h a v e a positive
then it is likely that 3 Ik.
This section is devoted to s o m e
tive binary quadratic f o r m s
special e x a m p l e s .
of discriminants
-Z700
sition g r o u p tables, respectively.
Discriminant
-Z700
C 1 = (i, 0,675)
C z = (Z5, 0, Z7)
and
We
list the r e d u c e d p r i m i -
-18ZbZ,
and their c o m p o -
30
C 3 = (13, 2, 52)
C 4 = (4, Z, 169)
C 5 = (7, 4, 97)
C 6 = (9,6,76)
C 7 = (19, 6, 36)
C 8 = (25, i0, 28)
C 9 = (ZS, Z0, 3t)
CI0 = (27, 18, ZS)
and, for
3 < n < i0,
Cn+8 = ~n"
Here,
if
C = (a,b,c),
then
= (a, - b , c).
{ T h e notation,
(a,b,c)
for the binary quadratic f o r m
ax
2
Z
+ bxy + cy , is a
standard one. }
G r o u p Table for D i s c r i m i n a n t
T h e generators are
C3
C I = C30
and
-2700
C7 .
C 6 = C72C32
3
C Z = C3
C7 = C7
C3 = C3
2
C8 = C7 C3
2
Z
C4 = C3
C 9 = C7C 3
C 5 = C72C33
a n d for any
n
CI0 = C 7 C 3
(l<n<18)
C n ~ n =CI.
In the case of discriminant
sition g r o u p
G,
-2700,
we
can define a character of the c o m p o -
by
3
X ( C 7 C 3) = e
tl
6
e
t2
31
where
0 <_ tI < 3, 0 i tz < 6,
F o r each
and e a c h choice of tl, tz gives a different character.
tl, tz w e h a v e b e e n able to d e t e r m i n e the "twist"
18
T ( s ) = ~ x ( C .J) Z ( s , Cj)
j=l
by c o m p a r i n g
T(s)
with triple products of the type:
H
2k
12
1
l-
=I x(mod k) p
pS
where
1
X p, k(a) =
if
X
if
is just a n ordinary Dirichlet character
extended over all rational p r i m e s
L - s e r i e s of K u m m e r
fields.
soluble
z~__i
e
and
x k - a(p)
p.
(rood k),
and the inner product is
T h e s e are nothing m o r e
[See D. M .
abelian L-functions," Israeli J. M a t h . ,
Goldfeld " A large sieve for a class of non1973.]
W e find that
10
Z(s,C I) + Z(s,C Z) + Z
Z(s, Cj) = Z{K(S)
j=3
where
K : O(J:~),
Z(s, CI) + Z(S, Cz) + ZZ(s,C3) + 2Z(s, C4)
- Z(s,C
5)-
- Z(s,C
7) - Z ( s , C
Z(s,C
6)
8)
- Z ( s , C 9) - Z ( s , Cl0)
{K(S)
~(s)
where
K = Q(~/5),
than A r t i n - H e c k e
32
Z ( s , C 1) + Z ( s , C 2) - Z ( s , C 3) - Z ( s , C 4)
- Z(s,C
5) - Z ( s , C
6) - Z ( s , C
7)
+ (ZZ(s, C8)+ZZ(s, C 9) - Z ( s , Ct0
~K(S)
~(s)
where
K = Q(~/2).
Z ( s , C1) + Z ( s , Cz) - Z ( s , C3) - Z ( s , C 4)
+ ZZ(s, C5) - Z ( s , C6) + ZZ(s, C 7)
- Z ( s , C 8) - Z ( s ,
C 9) - Z ( s , Clo)
~K(S)
~(s)
where
K = Q(~//Z0).
Z ( s , C 1) + Z ( s , CZ) - Z ( s , C 3) - Z ( s , C 4)
- Z ( s , C5) + ZZ(s, C 6) - Z ( s , C 7)
- Z(s,
C8) - Z(s,
=g"
where
C 9 ) + ZZ(s, C10)
~K(s)
~(s)
K = Q(~//10).
Z(s, CI) - Z(s, Cz) - 2Z(s, C3) + ZZ(s, C4)
+ Z(s, C5) - Z(s, C6) - Z(s, C7)
+ Z ( s , C8) - Z ( s , C9) + Z ( s , C10)
~K(S)
~(s)
where
K = Q(~//5).
T h i s l a s t c a s e ( i n v o l v i n g the 6th r o o t of 5) is m o s t i n t e r e s t i n g
and a l l o w s one
to e x t e n d D e d e k i n d ' s m e t h o d f o r o b t a i n i n g f o r m u l a e f o r the c l a s s - n u m b e r
of a f i e l d
33
Q(~/a)
to higher degree fields.
We
should like to cite the paper by H a r v e y C o h n
("A n u m b e r i c a l study of Dedekind's cubic class n u m b e r
formula, " Journal of l~esearch
of National B u r e a u of Standards, Vol. 59, 4, 1957, 265-271) w h e r e the c l a s s - n u m b e r s
of fields O(~/a)
are found for certain very large values of a.
W e would also like to record here that P r o f e s s o r Berndt has c o m m u n i c a t e d
to us that he has verified the non-vanishing of the functions
~K(S)
for fields
K = Q(~/a)
tioned above.
Serre
for a number
only recentiy
at
1
s = Z
of values
of
a,
found an example
using the type of formula
of a fieid
K
of degree
men8
for
which
~ K ( ~ ) = 0.
In the above
list of "twists"
we omitted
4 cases,
T h e s e are left as an exercise
to the reader.
Dis c riminant
-18Z5Z
C 1 : (I, 0,4563)
CI0 = (19,8,241)
C Z = (27, 0,169)
Cll
C 3 = (4, Z, 1141)
CIZ = (37, I0, IZ4)
C 4 = (7, Z, 65Z)
C13 = (163,1Z, 73)
C 5 = (Z8, Z, 163)
C14 = (Z7, 18, 17Z)
C 6 = (Z8, Z6,169)
C15 = (43, 18,108)
C 7 = (49, 44,103)
C16 = (36, 30,133)
c8
C17 = (67, 30, 76)
= (9, 6 , 5 0 8 )
C 9 = (36, 6, IZ7)
= (31,
i0,148)
C18 = (61, 3Z, 79)
C19 = (67, 46, 76)
Cn+17
= C
n
for
3 < n <
19.
G r o u p Table for Diseriminant
-18252
T h e r e are two generators, n a m e l y
C 4 = (7,2,652)
and
C 9 = (36,6,127)
34
T h e group table is as follows:
C4 z = C 7
3
C4
= C6
C44 = C 3
C 4 C 9 = C19
C 4 C 9 z = CIZ
Z
C 4 C 9 = C14
C 4 2 C 9 z = C18
C44C9
C44C9 Z = C 8
5
C4
= C16
3
= C5
C46 = C z
3
Z
C 4 C 9 = Cll
C4 C 9
= C17
C 4 5 C 9 = CIO
C 4 5 C 9 Z = C13
T o complete the table, use
C n " ~ n =C I .
A character
f o r t h i s g r o u p c a n be d e f i n e d a s f o l l o w s :
g~riatI
~
X (C4C 9) = e
lZ
Z~ri~tg
3
e
where
t1 : 0,1,...,11
t Z = 0, 1, Z.
T h e r e are four twists into H e c k e L-functions of pure cubic fields, n a m e l y
7
Z ( s , C1) + Z ( s , CZ) + Z z
j=3
Z(s, C)
J
~K(S)
19
Z ( s , cj)
j=8
where
=
2
•
-
-
~(s)
K = Q(~/13).
Z ( s , C 1) + Z(S, Cz) - Z ( s , C3) - Z ( s , C4) - Z ( s , C5)
+ Z Z ( s , c 6) - Z ( s , c 7) - Z ( s , c 8) + Z Z ( s , c 9) - Z ( s ,
clo)
+ ZZ(S, Cll ) - Z ( s , C12 ) - Z ( s , C13) - Z(S, Cl4) + Z Z ( s , C15)
- Z ( s , C16 ) + Z Z ( s , C17 ) - Z ( s , C18 ) - Z ( s , C19)
~K(S)
~(s)
35
where
K : O(~/Z).
Z(S, Cl) + Z(S, Cz) - Z(s, C3) - Z(s, C4) - Z(s, C5)
+ ZZ(s,06) - Z(s, C7) - Z(s,08) - Z(s, C9) - Z(s, CIo)
- Z(S, Cll) - Z(s, CIz) + ZZ(s, CI3 ) - Z(s, CI4) - Z(s, 015)
+ ZZ(s, CI6) - Z(s,017 ) + ZZ(s, C18 ) + ZZ(s, CI9 )
tK(S)
~(s)
where
§ 5.
K = Q(~/5Z).
In order to calculate the n u m b e r
of a given discriminant
t{ber Zahlentheorie,
A w e need to quote, for example,
f r o m Landauts V o r l e s u n g e n
Bd. 1.
Satz Z13. Jedes
und Z w a r
of primitive classes of binary quadratic f o r m s
ix - 0 oder
eindeuti~, = f m Z,
1 (rnod 4),
das Rein Q u a d r a t ist, is_t,
w___o m > 0 und
f Fundamental-diskriminante
ist.
Satz ZI4.
E s sei A = f m Z die zerlegung unseres
&
nach Satz 213.
D a n n ist
CO
K(a) : x (£)i:
r=l
r r
n
{i- (l)!]K(f).
p[m
P P
Here
~W
K(f) -
h(f).
See Satz Z09, pp. 15Z and 141. h(f) denotes the n u m b e r
binary quadratic f o r m s of discriminant
Using these theorems,
p
is a p r i m e
> 3,
of primitive classes of
f.
w e find that for the discriminant
the class n u m b e r
where
h(& ) is given by
f
h(-108 pZ)
Ix = -i08 p Z
J3(p-l)
if p-I(3)
1
if p ~ Z(3).
3(p+l)
Looking back at the linear combinations of Epstein's zeta functions
Z(s, Cj),
36
that o c c u r r e d in Section 4, it is natural to ask for the n u m b e r
Z(s, C.). This n u m b e r w h i c h is also the rank
J
zet{L functions of given discriminant
of linearly independent
p of the linear space of Epstein
Z
A = -I08 p ,
is given by
1
p = [ + ~ h(-108 p2).
So
p : i0 for
We
p : 5, p = 19 for
p =13.
This a g r e e s with our tables in §4.
note the following p h e n o m e n o n
the l i n e a r c o m b i n a t i o n s of the
which seems
w o r t h recording.
Z ( s , Cj) w h i c h a r e e q u a l to
2 •
Amongst
~K (s)
~(s-----~we o b t a i n
four combinations with
QI ' t,
o(b/pl,
Q (~.~'/Zp),
Q(~4p).
This has to do with the fact that the c o m p o s i t i o n g r o u p (of the primitive classes of
binary quadratic f o r m s with discriminant
index
3.
W h e t h e r the
K
w e w e r e unable to prove,
§ 6.
in
~K(s)
Z • ~(s)
-108 pZ)
has four distinct s u b g r o u p s of
is a l w a y s a sub-field of a K u m m e r
except in special cases above.
In o r d e r to p r o v e the a f o r e m e n t i o n e d assertions, w e develope s o m e
already m e n t i o n e d in a n earlier p a p e r (D. M .
Kurnrner field f2 = Q(~/I, k~/a), w h e r e
If p
field,
Goldfeld, loc. cit.). W e
a # + i or a perfect
is a rational p r i m e not dividing
ka
and
fl and
kth
ideas
consider the
power.
fz are m i n i m a l
such
that
p
then
p
fl _
= 1 ( n o d k),
x
k _ fz
= a
( m o d p)
soluble,
is u n r a m i f i e d a n d factors in i2 as a product of
k~ (k)
r
=
flfz
flfz
p r i m e ideals
local factor
~51,~Z ..... ~r
L
of
P
i.e.
(p) = i51~Z...~r
~[3(s)' w e see that
and
N~i = p
. L o o k i n g at the
37
L
/
lI / 1 -
:
P pip ~
1
=
_
N@)s
flfzs
P
Let
[l,~Z
be primitive
b
P
=
fl' fz th roots of unity, respectively.
fl
fz
]]
1]
hl=l hz=l
(
~
Then
Z
l - - -
Now define
~ 1 if x k ~a(p)soluble
XP'k(a) :
e Z1ri/k if x k ~ a(p) .
Then
-1
L
=
P
because, as
X
l-
runs through the Dirichlet characters
k1
value ~l exactly
value
i]
1]
x ( m ° d k) w=l
~ (k 1)
fl
times,
hz
@Z exactly k / f 2 t i m e s .
~(s)
=n
Also Xp, k(a)
( m o d k),
(w=l,Z
. . . . .
X(P)
takes on
k) a s s u m e s each
Consequently
L (s)
P
=
~v
P
-i
1]
L(s, X) II
1]
II
X (rood k)
w:l X (rood k) p
-
~'
p
1
Let
e o = Q ($'/t)'
K : Q(~,/a) .
T h e n following Artin (fiber die Zeta Funktionen y o n algebraischer Zahlkt~rper,
Collected Papers) w e get
1
~ ~0)~,
~,
38
Cons equently
n
=
~(s)
n
i
i Li
-
(6)
I w-I x(mod k) p
pS
k-
~7.
W e note that for discriminants
the positive integer n by
Cj
A = _3k Z,
the n u m b e r of representations of
(j fixed) is a linear combination of multiplicative
functions, which occur as coefficients in (6).
This is in accordance with Hecke theory (he, however, uses the H e c k e
operator, which does not appear in our work) according to which the n u m b e r of
representations of n > 0 by a quadratic f o r m in an even n u m b e r of variables > 4
is a linear combination of multip[icative arithmetic functions.
A s an example, if w e combine equations
(i) and
(4), and use the following
relation (here I< = Q(~/Z))
1
~K(s)
=
~(s)
[
z
1]
I] II
x ( m o d 3) w=l p
(
i-
x(p)×
P'
pS
3(z)
-i
(7)
co
:
m
O(n)
n=l n
s
which has been derived in Section 6, w e obtain
i~
z
xZ+z7yz(n) = ~
m (-i08)
E
kl n
I
l~4xZ+Zxy+TyZ(n) =-3Z
where
From
+ ZO(n
(8)
k
-I08
)]
kln~ (--~) - 0(n
i~
z(n) is the n u m b e r of ways n is representable by a x Z + b x y + cy Z.
axZ+bxy+cy
(7), it can easily be shown that 0(n) is a multiplicative function and defined
on the p r i m e powers
p
as follows:
39
-l,
f
0(p ~ ) =
where
(3
~)
derive
§ 8.
is the Legendre
many
equality between
finite number
relevant
stated
I
--" Z(3)
1,
~ --" 0(3)
In a s i m i l a r
(8)
series
(e. g. p r i m e s
it is also possible
of discriminant
fields).
with Euler
Z.
In t h i s p a r t
products
may
argument
say.
of the
exclude a
that divide the discriminant
With a special
to
- 3 k z,
of certain
these factors
can be
exactly.
With this convention,
(4)
gives
oo
3
Z~(s) 2 (-~)~ 1__=
1
using a well-known
Euier
fashion,
for forms
oroof at the end of Section
two Dirichlet
number
if x3 ~ Z(p)
I+(~)+... +(~)~ if x 3--z(p)
of the type
of Ubadn primes
algebraic
0,
symbol.
other formulae
We now give the promised
paper,
~ =-1(3)
product
theorem
n
of D i r i c h l e t .
for the left side of
(9)
n
s
~ Z(s,C.),
j=l
(9)
J
Here
(~---) i s t h e K r o n e c k e r
n
i s up to l o c a i f a c t o r s
(p=Z, 3)
symbol.
The
3
Z
II
( 1 - p - S ) - Z II ( 1 - p - Z S ) -1 = E Z ( s , C.) .
p---z(3)
For
the s a k e of clarity w e
j=l
p
(lO)
J
r e p e a t that
Z
C I = x
2
+ Z7y
C Z = 4 x Z + Z x y + 7y Z
C 3 = 4x 2 - 2xy + 7y Z
a n d that t h e s e
G's
earlier a c h a r a c t e r
form
a multiplicative g r o u p ,
o n this g r o u p
as r e c o g n i z e d
is defined as follows:
× ( c 1) = 1
× (C Z) =
Z
× ( C 3) = ~
.
by Gauss.
As noted
40
_i+v4"y
Here
~
=
- -
•
13
p~l(3)
a c u b e root of unity.
x(p)/_ z
1-
pS
]
p~ C 1
13
p-t(3)
We
now
(i - ll,3,n
pp
( 1 )_l
1 - --~s
p
(n)
pc C 3
pc C 2
13
p---Z(3)
introduct the "twist"~
3
= ~ x(Cj)Z(s,c.).
j=l
3
It is clear that
X(P) : l if p c G 1
X(P) = ~
if p c C Z
Z
X (P) = ~
We
observe
is that if p
if p c C 3
that the "twist" d o e s not t o u c h p r i m e s
is a p r i m e
The
r e a s o n for this
--- 2(3),
tions b y the totality of
G
representations
3
C.;
b y the
p =- 2(3).
a n d if the positive integer n h a s r(n) r e p r e s e n t a 2~
(j=l, 2,3), then n . p
h a s the s a m e n u m b e r ,
r(n), of
and, m o r e o v e r ,
there is a w e l l - k n o w n
I-i
correspond-
e n c e leading to
r(n.p
Since
C 2
o n the left side of
and
C 3
) = r(n),
if p --- 2(mod 3).
a r e "identical"
(ii) c a n b e c o m b i n e d ,
13
p-l(3)
1 -
1
13
p~l(3)
pc C 1
(in a s e n s e ~ )
1 -
a~
3
Y
j=l
We
recall equation (7):
×(c.)z(s,
3
a n d third factors
to give
pc C 2
=
the s e c o n d
c.).
3
1 - pS
13
1 p---Z(3)
(12)
41
~(s)
where
K = Q(3X//Z),
[
II
II
YI
× mod 3
w=l
p
(
1-
X(P)Xp'3(Z)
) p S
and
if x 3 ---Z(p) soluble
×p, 3(z) =
~izwi
e
3
if x 3 ~ 2(p) .
Comparing this with (12) w e get
~K(S)
3
z x(Cj)Z(s,C)j -- 2--~(s)
j=l
where
K = Q(~/2).
Equation
(i), up to local factors, follows, on noting that
x(Cz)Z(s, C 2) + x(C3)Z(s,C 3) = -Z(s, CZ).
§ 9. For the discriminant -1825Z, w e omitted in Section 4 to give the "twist" for
the case Q(~/Z6).
Z(s,C I)+ Z(s, Cz) - Z(s,C 3) - Z(s,C 4) - Z(s,C 5)
+ ZZ(s, C6) - Z(s, C7) + ZZ(S, Cs) - Z(s, C9) + ZZ(s, CI0)
- Z(s, CII ) + ZZ(s, CIZ) - Z(s, C13) + ZZ(s, C14) - Z(s, CI5)
- Z(s, CI6) - Z(s, CI7) - Z(s, CI8) - Z(s, CI9)
~K(S)
with
31
K = W(</26).
The following question is suggested by the above work:
combinations of Epstein zeta functions of a given discriminant
s a m e functional equation) have Euler products?
prime
> 3) w e found
In the case
H o w m a n y linear
A
(i. e. , with the
A = -108 pZ (p, a
42
1 + h(-a)
Z
such combinations.
T h e s e appear, to us, to be the only ones.
Finally, it is fitting to mention that the importance of the concept of twisting
w a s first recognized by A. Weil in his paper "Bestirnmung der Dirichletschen
Reihen durch ihre Funktional-Gleichungen"
(Math. Annalen,
1969).
Problems
and results on combinatorial n u m b e r
theory III
Paul ErdSs
Like the two previous p a p e r s of the s a m e
II) I will discuss p r o b l e m s
in n u m b e r
title (I will refer to t h e m as I a n d
theory w h i c h h a v e a combinatorial flavor.
T o avoid repetitions a n d to shorten the p a p e r as m u c h
previous results w h e n e v e r
convenient and will state as m a n y
possible, and will discuss the old p r o b l e m s
some
as possible I will refer to
new problems
as
only w h e n they w e r e neglected or if
n e w result has b e e n obtained.
P. E r d ~ s ,
Problems
and results on combinatorial n u m b e r
theory I and II,
a s u r v e y of combinatorial theory, 1973, N o r t h Holland, I17-138; J o u r n 4 e s Arithm & t i q u e s de B o r d e a u x
papers have many
hombres,
Juin 1974, A s t 4 r i s q u e Nos.
references.
Monographies
(1963), 81-135.
Graham
Z4-Z5,
Z95-310.
See also Q u e l q u e s p r o b l ~ m e s
de i' E n s e i g n e m e n t
Math&matique
Both of these
de la th4orie des
iNo. 6, Univ. de G e n e v a
and I will soon publish a p a p e r w h i c h brings this p a p e r up
to date.
P. E r d B s , S o m e
a n d Publ. Math.
I.
unsolved p r o b l e m s ,
Inst. H u n g a r .
Acad.
M i c h i g a n Math.
J. 4(1957), Z91-300
Sci. 6(1961), ZZI-Z54.
First I discuss V a n der W a e r d e n ' s and S z e m e r ~ d i ' s t h e o r e m
tions.
D e n o t e by
exceeding
and related ques-
f(n) the smallest integer so that if w e divide the integers not
n into two classes then as least one of t h e m
p r o g r e s s i o n of n t e r m s .
More
generally, denote by
contains an arithmetic
f (n) the largest integer so
u
that w e can divide the integers not exceeding
every arithmetic p r o g r e s s i o n of n
f (n) into two classes so that in
u
n+u
e a c h class has f e w e r than --7- terms.
terms
T h e best l o w e r b o u n d for f(n) is due to B e r l e k a m p ,
(f(p) > pZP
if p
is a p r i m e a n d
to decide if f(n)I/n -- o0 is true.
f(n) > cZ n
My
for all n).
L o v ~ s z and myself,
It w o u l d be v e r y interesting
g u e s s w o u l d be that it is true.
I p r o v e d by the
probabilistic m e t h o d that fu (n) > (l+E c )n if u > cn. T h e proof gives nothing if u
/
is 0(nl/Z). It w o u l d be v e r y interesting to give s o m e usable u p p e r a n d l o w e r
b o u n d s for
f (n). A s far as I k n o w
(Bull. Canad.u Math.
fz(n)
the only result is due to J. S p e n c e r w h o p r o v e d
Soc. 16(1973), 464)
fl(n) = n(n-l),
equality only if n = zt"
is not k n o w n .
F o r various other generalizations (see II).
D e n o t e by
rk(n) the smallest integer so that every s e q u e n c e
1 <_a I < ... < a~ <_n, ~ = rk(n ) contains a n arithmetic p r o g r e s s i o n of k
terms.
44
Szemer6"di r e c e n t l y p r o v e d the old c o n j e c t u r e of T u r i n and m y s e l f
(1)
rk(n) = 0(n).
(i) of c o u r s e contains V a n d e r W a e r d e n ' s t h e o r e m .
for
r3(n ) are due to B e h r e n d
best
The
estimates
and IKoth w h o p r o v e d
n
c n
Z
r3(n) < loglog n
<
Cl(lOg n) I/Z
e
T h e true o r d e r of m a g n i t u d e
of
rk(n)
is v e r y difficult to d e t e r m i n e .
I w o u l d ex-
pect that
rk(n)
=
0
(log n) ~'
holds for e v e r y
k
and
~ . A v e r y attractive conjecture of m i n e
states: Let
1
l < a I < a Z < ... be an arbitrary s e q u e n c e of integers with ~ - - = co . T h e n our
a.
i
s e q u e n c e contains for e v e r y k an arithmetic p r o g r e s s i o n of k t e r m s .
I offer
3 0 0 0 dollars for a proof o r d i s p r o o f of this conjecture.
w o u l d i m p l y that for e v e r y
There
k
there are
k
is an interesting finite f o r m
primes
is extended o v e r M 1
contain an arithmetic p r o g r e s s i o n of k
G
< 0o,
in fact p e r h a p s a l r e a d y
u p p e r a n d l o w e r b o u n d s for
A k.
sequences
terms.
A 3 = o0,
I am
w h i c h do not
it w o u l d be v e r y desirable to h a v e g o o d
afraid u p p e r b o u n d s are h o p e l e s s at p r e s e n t I o b s e r v e d that A k >
k io~ Z
2
recently p r o v e d
(Z)
A k > (l+o-(l))k log k .
His proof will s o o n a p p e a r in Proc.
may
a I < a Z < ...
It is not at all obvious that
so one should p e r h a p s concentrate on l o w e r bounds.
and Gerver
Put
I
a.
l
i
the m a x i m u m
if true,
in arithmetic progression.
of our conjecture:
A k : max
where
T h e conjecture,
Amer.
Math.
Soc.
Gerver
believes that
be best possible.
Perhaps
the following t w o further functions a r e of s o m e
interest:
Put
(Z)
45
1
a.
Ak,n,~ ~ = max
a .<n
i
1
w h e r e the m a x i m u m
is to be taken over all s e q u e n c e s w h i c h do not contain a n arith-
m e t i c p r o g r e s s i o n of k
terms.
I of c o u r s e expect
interest to estimate f r o m a b o v e and b e l o w
. (n) = m a x
/~k
where
the
for e v e r y
fixed
and
n
a.
Ak(n) < ck.
A k - Ak(n).
K
a.>n
i
It m i g h t be of
Define next
1
a.
1
do not contain an arithmetic p r o g r e s s i o n of k
terms.
i
I expect that
A (n)
for
k l i m a (n) = 0. It m i g h t be of interest to investigate A k - ~ k
k
n=o0
and large k. In particular is it 0(log n)? A l s o w h a t h a p p e n s if both k
n tend to infinity?
It is not clear w h i c h if any of these questions will lead to
fruitful results.
A s e q u e n c e of integers
l < a I < ... < a k <_n
a. is fine arithmetic m e a n of other a' s.
1
T h e study of these s e q u e n c e s w a s
We
is called n o n - a v e r a g i n g if no
started by E. Straus.
Put
maxk
= g(n).
have
(3)
e c(l°g n)I/Z < g(n) < n Z/3+E
T h e l o w e r b o u n d in (3) is due to Straus the u p p e r b o u n d to Straus a n d myself.
7
A b b o t t recently p r o v e d the u n e x p e c t e d
to d e t e r m i n e
Very
g(n) > cn I/I0.
It w o u l d be v e r y interesting
lira log g(n)/log n.
n=o0
recently F u r s t e n b e r g p r o v e d S z e m e r ~ d i ' s t h e o r e m
by m e t h o d s
of
ergodic theory, his proof will be published soon.
References
I~. S z e m e r 6 d i ,
progression,
of the p r o b l e m
O n sets of integers containing no
A c t a Arith. Z7(1975), Z99-345.
e l e m e n t s in arithmetic
F o r further literature a n d history
see I; II a n d the p a p e r of S z e m e r e d i .
]2. G. Straus, N o n a v e r a g i n g
sets, P r o c .
Syrup. P u r e Math.
P. E r d B s a n d E. G. Straus, N o n a v e r a g i n g
Math.
k
AMS
sets II, Coll. Math.
1967.
B61yai
Soc. C o m b i n a t o r i a l theory a n d its applications, N o r t h Holland A m s t e r d a m -
L o n d o n 1970 Vol. Z, 405-411.
46
Z.
Covering
A system
congruences
a n d related questions.
of c o n g r u e n c e s
(1)
al (rood n.l)'
is called a covering
system
l<nl<" " " < n k
if e v e r y integer satisfies at least one of the c o n g r u e n c e s
(i). T h e principal conjecture w h i c h is n o w m o r e
can be arbitrarily large.
It is surprising h o w
500 dollars for a proof or disproof.
system
with
n I = 20.
than 40 y e a r s old states that
The
difficult this conjecture is -- I offer
r e c o r d is still held by Choi w h o
= min >
___I
n
n.
nl=n
the m i n i m u m
gives a
Put
u
where
n1
1
is to be taken o v e r all covering
systems
a . ( m o d n.).
1
I conjecture
1
that
(Z)
u
n
-oo
as
n-oo.
If (Z) is true it w o u l d be interesting to estimate
u
from
a b o v e a n d below.
n
Put
f(n) = m i n k
vchere the m i n i m u m
perhaps
F(n) = rain a k
is extended o v e r all s y s t e m s
interesting to get non-trivial b o u n d s for
Here
and
it is w o r t h w h i l e
(I) with
f(n) a n d
n I = n.
It w o u l d be v e r y
F(n).
to introduce a n e w p a r a m e t e r .
u (c) = m i n >
n
1
n.
Put
n[ = n
'
1
where
the m i n i m u m
is extended o v e r all finite s y s t e m s
a . ( m o d n.),
n = n I <n2<
for w h i c h the density of integers not satisfying a n y of these c o n g r u e n c e s
than or equal to
c,
Estimate
or d e t e r m i n e
the a s y m p t o t i c
properties
...
is less
of u (c) as
n
c -- 0
and
n-- ~ .
Similar questions
I conjecture that for every
n.1 a r e s q u a r e - f r e e
£
integers all w h o s e
can be a s k e d about
f(n, c) a n d
there is a covering s y s t e m
prime
F(n,c).
(1) w h e r e
factors a r e greater than
p~ .
all the
47
Let
n I < n Z < ... < n k
ing to obtain conditions
system
(I) exists.
0-(n)/n > C,
be a s e q u e n c e
of moduli.
(if possible n e c e s s a r y
In particular
but no s y s t e m
It w o u l d be v e r y interest-
a n d sufficient ones) that a covering
I conjecture that for every
(I) exists w h e r e
the
C
there is an
n
with
n. > 1 are the divisors of n.
On
1
the other h a n d B e n k o s k i a n d I conjectured that if cr(n)/n > C
distinct p r o p e r divisors of n.
smallest value of
C
then
n
is the s u m
of
If this conjecture is true w e w a n t to estimate the
for w h i c h the conjecture holds.
A n older conjecture of B e n k o s k i
states:
if n
is odd a n d
0-(n_._._~)> Z then
n
n
is the s u m
of distinct p r o p e r divisors of
n.
O n e can also study infinite covering
his students but to avoid trivialities
satisfy a c o n g r u e n c e
m
gruences
if k > k0(E)
i
--
done by Selfridge a n d
every
m
> m 0
must
A n o t h e r possibility w o u l d be to
1
the density of the integers satisfying none of the con-
a . ( m o d n.) 1 < i < k
I
as w a s
one usually insists:
~ a . ( m o d n.), n~ > n .
I
require that
systems
is less than
E .
Perhaps
the first condition implies
I
the second.
D e n o t e by
P1
N
the s e q u e n c e
I < nl<
if for e v e r y choice of residues
nz<
...
of rnoduli.
a . ( m o d n.) a n d to e v e r y
1
N
e > 0
h a s property
there is a
k
1
so that the density of integers satisfying n o n e of the c o n g r u e n c e s
(3)
a.(mod
n.)
1
is less than
E .
N
1< i< k
1
is said to h a v e property
PZ
if there is a s e q u e n c e
of residues
a. so that the density of the integers satisfying n o n e of the c o n g r u e n c e s (3) is
i
k
less than a . It has p r o p e r t y P 3 if this holds for a l m o s t all (i. e. 0( II ni) )
i=l
choices of the residues a.. P 3 clearly holds if there is a s u b s e q u e n c e
{nir } with
1
1
r ni r = 0o, (nirl, n irZ) = i, but at the m o m e n t
I do not see a n e c e s s a r y a n d sufficient
condition.
P2
certainly
d o u b t h o i d s if a n d o n l y if
these
iines and must
trivial
1
1
is
equivalent
of integers
O.
but it also holds if
I formulated
these problems
of the reader
to choose
with the condition:
which does not satisfy
On the other hand observe
so that every integer
suffices
1
iIE ~-i = ~ "
ask the indulgence
is clearly
the density
1< k < ~
a.
ia~.~-=l~i = ~ ,
if s o m e
n.1 = Zi"
Pl
no
whiie writing
of t h e q u e s t i o n s
are
or false.
Pl
a.
h o i d s if
a. = i.
1
satisfies
For
every
choice of the residues
any of the congruences
that it is trivial
1
that one can find residues
at least one of the congruences
By a s l i g h t m o d i f i c a t i o n
ai(mod n.),
a.(mod
1
n.) - -
we can obtain a problem
1
which
48
is p e r h a p s
not trivial:
Let
n I < n Z < . . . w h a t is the n e c e s s a r y
condition that residues
a. exist so that all but a finite n u m b e r
i
satisfy one of the c o n g r u e n c e s
(4)
m
O n e can also ask:
--- a . ( m o d
1
W h a t is the n e c e s s a r y
integers satisfy one of the c o n g r u e n c e s
n.),
1
m
>
a n d sufficient
of integers
m
n..
1
a n d sufficient condition that a l m o s t
all
(4)?
F o r particular choices of the
decide if a l m o s t all integers
a. (say a. = 0) it often is v e r y h a r d to
i
1
satisfy one of the c o n g r u e n c e s
a.(modl ni)" A v e r y
old p r o b l e m
Is it true that a l m o s t all integers h a v e t w o divisors
of m i n e
states:
d I < d z < Zd I.
If this conjecture is correct one could c h o o s e as m o d u l i the integers w h i c h
are minimal
relative to the property of having t w o divisors
in the s e n s e that no p r o p e r divisor has that property.
determine
Many
dI < d2<
Zd l
T h e choice
a set satisfying at least one of the c o n g r u e n c e s
a n d density
di, d 2 with
a. = 0 w o u l d then
i
with infinite c o m p l e m e n t
I.
further questions c a n be a s k e d but I leave their formulation
to the
reader.
A set of c o n g r u e n c e s
integer satisfies at m o s t
system
a i ( m o d ni) , n I < n 2 < ...
one of these c o n g r u e n c e s .
is called disjoint if every
I conjectured that no covering
can be exact i. e. every integer satisfies exactly one of the covering
congruences.
Mirsky
and Newman
v e r y s i m p l e proof of m y
Let
a i ( m o d hi),
be a disjoint system.
as possible.
Put
Szemeredi
a n d R a d o found a
conjecture.
Stein a n d I asked:
(5)
a n d a little later D a v e n p o r t
i < n I < ... < n k < x
maxk
= g(x),
determine
or estimate
g(x)
as accurately
and Iproved
m1+ E
-Cl(lOg x)
xe
Z
<
g(x)
<
x
c2
(log x)
We
believe that the l o w e r b o u n d is closer to the truth.
Szemer6di
and I tried
49
unsuccessfully to give n e c e s s a r y and sufficient conditions for a s e q u e n c e of m o d u l i
n I < ... < n k
that a disjoint s y s t e m
A s far as I k n o w
a.(modl n.),1 1 < i < k
the following question w h i c h m a y
be of s o m e
interest has
not yet b e e n investigated:
L. for w h i c h (5) is a disjoint s y s t e m be
greater than
obably
m.
cm
L e t all th~
1
= m a x Z n.
should exist.
1
em
--0
as
m--
oo. If true estimate
1
. Perhaps
greater than
it w o u l d be better to require that all p r i m e factors of the
covering congruences.
sizes.
are
m.
T h e r e are m a n y
SchBnheim:
n
Let
~
recent generalisations of covering c o n g r u e n c e s and exact
H e r e I only state a beautiful conjecture of H e r z o g a n d
kbe a finite A b e l i a n group.
HI,...,H. k
are cosets of different
P r o v e that i=~ H i n e v e r gives an exact covering of ~.
References
Summa
P. E r d ~ s , O n the integers of the f o r m
Brasil Math. ii(1950), I13-IZ3.
P. E r d ~ s a n d I<. S z e m e r ~ d i ,
A r i t h m e t i c a 15(1968), 85-90.
P. E r d ~ s , O n a p r o b l e m
L a p o k 3(195Z), IZZ-IZ8.
Zk + p
On a problem
on s y s t e m s
and s o m e
related p r o b l e m s ,
of E r d ~ s and Stein, A c t a
of c o n g r u e n c e s
(in H u n g a r i a n ) Mat.
S. L. G. Choi, C o v e r i n g the set of integers by c o n g r u e n c e classes of
distinct moduli, Math. C o m p . Z5(1971), 885-895.
Comp.
S. J. B e n k o s k i and P. E r d ~ s ,
Z8(1974), 617-623.
O n w e i r d a n d pseudoperfect n u m b e r s ,
Math.
50
3.
Some
applications
of covering c o n g r u e n c e s
In 1934 i%omanoff p r o v e d that the l o w e r density of integers of the f o r m
2k + p
many
is positive.
He wrote me
w h e t h e r I can p r o v e that there are infinitely
o d d integers not of the f o r m
on covering c o n g r u e n c e s
2k + p.
and I proved
there is an arithmetic p r o g r e s s i o n
w h i c h is of the f o r m
2k + p
This question led m e
to the p r o b l e m s
-- using covering c o n g r u e n c e s
-- that
consisting entirely of odd n u m b e r s
(this w a s p r o v e d independently
no t e r m
of
by V a n d e r C o r p u t
too).
It is easy to see that if one could p r o v e that there a r e covering
w i t h arbitrarily large
metic progression
most
r
n I then it w o u l d follow that for e v e r y
no t e r m
distinct p r i m e
of w h i c h is of the f o r m
T h e following question s e e m s
of
27
the f o r m
where
r
O
r
It is e x t r e m e l y
and
o d d integers not of
recently p r o v e d
is s q u a r e f r e e ?
~ > 0 there is a n r > r0(~ ) so that the l o w e r
kI
kr
+ ... + 2
is greater than I - ~ .
Let
PI' " " " " Pk
becomes
be the s e q u e n c e
f(n) the n u m b e r
of integers with
of solutions of
f(n) > 0.
h o p e that the density of our s e q u e n c e
of this type s e e m s
that
Zk + L
where
Is it true
Z
Pi ~ L
a n d let a I < a 2 < ...
In v i e w of i%omanoff's result one w o u l d
exists.
to be far b e y o n d our resources.
n,f(n) > o log log n
105
~ai}
2k + p = n
Unfortunately to decide questions
I p r o v e d that for infinitely
but could not decide w h e t h e r
is the largest integer for w h i c h all the n u m b e r s
f(n) = 0(log n).
primes.
I am
fairly certain that this conjecture is true.
likely that for infinitely m a n y
squarefree.
n
all the integers
I conjectured
n - 2k, 1 < k < log n
--
seems
for
i = l,...,k?
D e n o t e by
many
a p p a r e n t if w e p o s e
be a n y finite set of p r i m e s .
that e v e r y sufficiently large odd integer is of the f o r m
every
2k + @
Is there in fact a n odd integer not of this f o r m ?
T h e connection with covering c o n g r u e n c e s
the following question:
(using
p + 2
Is it true that e v e r y sufficiently large o d d integer is of the f o r m
@
r
or f e w e r
doubtful if covering c o n g r u e n c e s
In the opposite direction Gallagher
of Linnik) that to every
r
D o they contain an
Schinzel p r o v e d that there a r e infinitely m a n y
density of integers of the f o r m
where
has at
Is it true that for every
of a p r i m e
Is the density of these integers positive?
p + Zk + Z £.
the m e t h o d
v e r y difficult:
o d d integers not the s u m
infinite arithmetic p r o g r e s s i o n ?
will help here.
r there is an arith-
factors.
there are infinitely m a n y
powers
2k + O
systems
log
Z
O n the other h a n d it
n - 2k '
Zk < n
are
are
51
Incidentally I a m
sure that lira (ai+1 - a.) = oo.
This would certainly
follow if there are covering s y s t e m s with arbitrarily large
n I.
T h e following s o m e w h a t v a g u e conjecture can be formulated.
Consider all
the arithmetic progressions (of odd n u m b e r s ) no t e r m of w h i c h is of the f o r m
2k + p.
Is it true that all these progressions can be obtained f r o m covering
congruences and that all (perhaps with a finite n u m b e r
in any of these progressions are of the f o r m
of exceptions) integers not
Zk + p?
Finally C o h e n and Selfridge proved by covering congruences that there is
a n arithmetic progression of odd n u m b e r s
no t e r m of w h i c h is of the f o r m
Zk + p ~
and Schinzel used covering congruences for the study of irreducibility of polynomials.
References
P. ErdSs, O n integers of the f o r m Zk + p and s o m e related problems,
Summa
Brasil Math. 2(1950), 113-123. F o r further literature on covering congruences
see P. ErdSs, S o m e p r o b l e m s in n u m b e r theory, C o m p u t e r s in n u m b e r theory,
Proc. Atlas Syrup. Oxford 1969 Acad. P r e s s 1971, 405-414.
A. Schinzel, Reducibility of polynomials, ibid. 73-75.
F. C o h e n and J. L. Selfridge, Not every n u m b e r is the s u m or difference
of two p r i m e powers, Math. of C o m p u t a t i o n Z9(1975), 79-8Z.
52
4.
An
if n o
1
~--<
ai
Some
unconventional
infinite s e q u e n c e
extremal
I_< a I < ...
problems
of i n t e g e r s is called a n
A
sequence
a. is the distinct s u m of o t h e r a's. I 0 r o v e d that for e v e r y A
sequence
I
1
i00. Sullivan o b t a i n e d a v e r y substantial i m p r o v e m e n t ,
he proved
~--<
ai
It w o u l d
b e interesting to d e t e r m i n e
z!
max
where
the m a x i m u m
4.
is e x t e n d e d
ai
o v e r all A
sequences.
greater than
Z.
b I < b Z < ...
some
so that t h e r e s h o u l d b e a n
absolute constant
the o t h e r s e q u e n c e s
Perhaps
A
Sullivan c o n j e c t u r e s that this m a x i m u m
Is it p o s s i b l e to obtain n e c e s s a r y
sequences
c
and every
considered
A
n v
sequence
The
the inequalities of L e v i n e
(see their f o r t h c o m i n g
E 1
ai
shows
< log Z + ~
that this is best
Usually
and
one
the
is rarely
Here
mentioned
Another
Let
as
As
I r e f e r to this p r o b l e m
as
(I).
(I) for
is a n
t e n d s to infinity,
npn
+
A
sequence
I,...
,
Zn
of these
extremal
and
of i n t e g e r s
Ryavec
problems
and
others
s u c h that all the s u m s
is difficult
proved
n
i
that if
~iai '~ i =0
n
~ -!-I < Z - zn_----l 7- equality if a n d only if a. = zi-l.
-i
i=l
ai
oldest p r o b l e m s
i _ < a I < ... < a n _ < x
Is it true that
far as I k n o w
c o u p l e of s i m p l e
i< a I <
q u e s t i o n c a n b e a s k e d for all
that if n_< a I < ...
n
I conjectured
II): L e t
a r e distinct.
follows:
--0
determination
is a s e q u e n c e
p r o o f o r disproof.
for
possible.
exact
I and
Cbn
p a p e r in A c t a A r i t h m e t i e a ) .
I call attention to o n e of m y
in
an <
n
1 a r e all distinct t h e n
n
i~=l~iai
=
E
successful.
1 _< a.l < " " " < a n
or
where
n
satisfying
a n d Sullivan c a n solve p r o b l e m
I c o n j e c t u r e d a n d L e v i n e just p r o v e d
then
same
in this p a 0 e r .
is only a little
a n d sufficient conditions for
b e s u c h that all the s u m s
n < log x + C ?
log Z
I offer 3 0 0 dollars for a
this c o u l d h o l d w i t h
extremal
( w h i c h is of c o u r s e
problems
C = 3.
w h i c h I c a n not s o l v e state a s
...
b e a s e q u e n c e of i n t e g e r s for w h i c h all the s u m s
a + a.
-i
j
1
a r e different. D e t e r m i n e
max )2--.
W e get different p r o b l e m s if i = j is
ai
p e r m i t e d o r not -- but I c a n not s o l v e a n y of t h e m .
Let
a's.
a 0 = 0, a I = 1 < a Z < ...
i
Determine
rain ~ - - .
ai
In s o m e c a s e s o n e e n c o u n t e r s
our sequence
let
sum
has density
a I < a Z < ...
of t w o g r e a t e r
0
b e s u c h that e v e r y i n t e g e r is the s u m
problems
but it is m u c h
b e a n infinite s e q u e n c e
a's.
of t w o
where
it is not h a r d to p r o v e that
I
~-< 0%
e.g.
i ai
of i n t e g e r s w h e r e n o a.i divides the
harder
S~rkozi and I proved
to p r o v e that
that the d e n s i t y of s u c h a s e q u e n c e
53
is 0 but w e could not prove
(I).
The
following
such
that
no
say,
if
finite
a. divides
1
x = 3n and the
1
~--<
a.
i
problem
o0
and are n o w h e r e near of settling p r o b l e m
remains
the
sum
of two
a's
are
the
here.
greater
integers
Let
a's.
1 <__ a 1 < . . . < a k <__ x
Then
Z n , Z n + 1. . . .
k < [ 3 ] + 1.
-,3n.
be
Equality,
References
P. Erdbs, P r o b l e m s and results in additive n u m b e r theory, Colloque sur
la th~orie des n o m b r e s , Bruxelles. G e o r g e Thone, Li@ge; M a s s o n and Cie,
Paris (1955), IZ7-137.
P. E r d B s and A. S~rkozi, O n the divisibility properties of sequences of
integers, Proc. L o n d o n Math. Soc., ZI(1970), 97-101.
54
5.
Some
more
extremal
problems
in additive a n d multiplicative n u m b e r
theory
S i d o n calls a s e q u e n c e of integers I <__a I < . . . a B k s e q u e n c e if the s u m s
k
~.~ a
,s . = 0 or 1 a r e all distinct. S i d o n a s k e d in 1933: find B k s e q u e n c e for
i=l i r i
I
which
a
tends to infinity as s l o w l y as possible.
It is e a s y to see that t h e r e is
n
a B Z s e q u e n c e with a n <
C n 3 for all n. O n e of the m o s t challenging p r o b l e m s
here
states:
Is t h e r e a
B Z
sequence
with
a / n 3 -- 0 -- I give i00 dollars for a
n
p r o o f o r disproof.
I of c o u r s e
e x p e c t that s u c h a s e q u e n c e exists -- in fact I a m
Z+~
s u r e that t h e r e is a B Z s e q u e n c e with a < n
for e v e r y E > 0 a n d n > n0(s ).
n
2+~
P~gnyi a n d I p r o v e d b y probabilistic m e t h o d s that t h e r e is a s e q u e n c e
a < n
n
for w h i c h
f(m) =
~
1 < c
ai+aj=m
~
the o t h e r h a n d I p r o v e d
where
the c o n s t a n t
c
depends
that for e v e r y
B 2
sequence
n
= oo in fact
if { a n }
is a basis t h e n
An
n
l i m sup n=o0
n
n
(i) be i m p r o v e d ?
On
a
lim sup-~
Can
E .
E
a
(i)
only o n
> 0.
log n
old c o n j e c t u r e of T u r i n
lira s u p f(n) = co,
more
and myself
generally:
let a
states that
Z
< c n D
n
n = l,Z,..,
is it then true that
lira s u p f(n) = 0o ?
I offered a n d offer 3 0 0 dollars
for a p r o o f or d i s p r o o f of these c o n j e c t u r e s .
The
interval.
exceeding
B
sequences
Denote
n.
behave
quite differently if w e
by
Bk(n)
the m a x i m u m
Turin
and I
(see also C h o w l a )
(l+o(1))n I/Z <
(l)
(i) is of c o u r s e
mentioned
in
I and
number
restrict t h e m
of t e r m s
of a
Bk
to a finite
sequence
proved
Bg(n) < n I/2 + c n I/4 .
II.
We
conjecture
B z ( n ) = nl/Z + o(i) .
(Z)
I offer 3 0 0 dollars for a p r o o f o r d i s p r o o f of (2).
Bose
a b o v e for
with Turan
and Chowla
k >__3.
breaks
They
observed
observe
down
and
that it is v e r y h a r d to e s t i m a t e
B 3 ( n ) >__ (l+o(1))n I/3
B 3 ( n ) <__ (l+o(1))n I/3
Incidentally if a I < a Z < . . .
that
lira s u p a n / n 3
= oo ,
but r e m a r k
Bk(n)
from
that o u r p r o o f
is open.
is a n infinite
B3
sequence
t h o u g h I h a v e no d o u b t that this is true.
I cannot prove
not
55
We
a r e v e r y far f r o m
as far as I k n o w
which
being able to solve p r o b l e m
there is not e v e n a reasonable
(I) for
Bk
sequences
--
conjecture.
L e v i n e in a recent letter to m e asked: Is there a B Z s e q u e n c e a n d a n
1
~
i/2
~ _ o0 o It follows f r o m m y results that for every 6 > 1
i ai
(log ai)
s >
l
(3)
~
ai)i/Z(log
)6
i (a i l o g
l o g a.
1
But I h a v e an e x a m p l e
of a
B 2
< o0.
s e q u e n c e for w h i c h
1
(4)
z
I/z
i a.
1
=
~"
(log log a.)
1
In trying to close the gap b e t w e e n
(3) a n d
(4) L e v i n e asked:
Is it true
that
(ai log ai)i/Z(log log a.)1
converges
for every
I proved:
BZ
sequence?
There
exists a n infinite
BZ
s e q u e n c e with
Bz(n)
(5)
lira s u p
n=0o
i/2
in
(5) w a s
improved
possible result could be
to I/Z I/Z
i.
by K r u c k e b e r g .
a I < a Z < ... < a k
modulus
a n d a perfect difference set
There
is an integer
b l , . . . , b u + 1 so that every residue
the
a's
occur amongst
the
be a n y
m
= u
mod
BZ
rood m
Z
sequence.
m
T h e n there exists a
w h i c h contains the
+ u + 1 and
seem
like this h a v e b e e n investigated a n d in s o m e
structures.
(I) the best
u + 1 residues
is uniquely of the f o r m
a' s.
In
rood m ,
b.1 - b.j a n d
b's.
This conjecture if true w o u l d
a n d others,
In v i e w of
This w o u l d follow if the following conjecture of m i n e
Let
m
,
n
w o u l d hold:
other w o r d s :
1
--Ui-/z _>~
for Steiner s y s t e m s
to m e
to be v e r y interesting.
cases solved by T r e a s h ,
a n d other m o r e
Questions
Lindner
complicated combinatorial
0
56
References
P. ErdBs and A. P~nyi, Additive properties of r a n d o m sequences of positive
integers, Acta Arith 6(1960), 83-110, see also Halberstam-P~oth, Sequences, Oxford
Univ. Press, 1966.
P. ErdSs and P. TurAn, O n a p r o b l e m of Sidon in additive n u m b e r theory
and on s o m e related problems, Journal L o n d o n Math. Soc. 16(1941), Z12-Z16, A d d e n d u m 19(1944), 208.
A. Stbhr, Gelbste und ungelbste F r a g e n ~iber B a s e n der nat~irlichen Zahlenreihe
I, II J. reine a n g e w Math. 194(1955), 40-65, ili-140. This paper has m a n y
p r o b l e m s and results and a very extensive bibliography. It contains the proof of
(i) and (5).
C. Treash, T h e completion of finite incomplete Steiner triple systems
with applications to loop theory, J. Combinatorial Theory, Set A 10(1971), Z59Z65, for a sharper result C. C. Lindner, E m b e d d i n g partial Steiner triple systems,
ibid 18(1975), 349-351.
Math.
F. Kr[{ekeberg, B z - F o l g e n und verwandte Zahlenfolgen, J. reine a n g e w
Z06(1961), 53-60.
57
6.
Graham
Problems
on infinite subsets
a n d Rothschild conjectured that if w e
classes then there a l w a y s is a n infinite s e q u e n c e
split the integers into t w o
aI < a 2 < ...
so that all the
finite s u m s
(i)
52ekak ,
a r e in the s a m e
ek = 0
or
1 .
class.
This conjecture w a s
fied by B a u m g a r t n e r .
proved
recently by H i n d m a n
a n d the proof w a s
simpli-
I just h e a r d that G l a s e r using a n idea of Galvin obtained a
v e r y interesting topological proof of the t h e o r e m .
A f e w days a g o I asked:
Is there a function
f(n) so that if w e
integers into t w o classes there a l w a y s is a s e q u e n c e
holds for infinitely m a n y
n
Galvin just s h o w e d
splitting as follows:
Let
a n d so that
that no s u c h
F(m)
in the first class if y > F ( x )
-- o0
a I < ...
split the
for w h i c h
an < f(n)
(I) holds ?
f(n) exists.
T o see this he defines a
x
sufficiently fast. P u t n = g y, y odd. n
a n d is in the s e c o n d class if y < F(x).
is
It is easy
to see that this construction gives a counter e x a m p l e .
There
problem.
k
(or
m i g h t be t w o w a y s
to save the situation a n d obtain s o m e
Is it true that there is a n
N0)
classes there is a s e q u e n c e
one of the classes is disjoint f r o m
ask a weaker
classes,
statement:
i.e.
{A
}
The
ekXk
{x }
of p o w e r
c
a n d the s u m s
x + y = z
of S ?
S, y ~
S
sequence
of S
a l m o s t disjoint
and
co
c
{an},
all the classes
is
A
into t w o classes.
so that all the s u m s
Let
Sx
be a set of real
Is there then a set
so that all the s u m s
If the a n s w e r
is no then w e
{Xc~1 + x a 2 }
also
could p e r h a p s a s s u m e
a r e distinct.
V a n d e r !~aerden's t h e o r e m .
of positive density.
n
is not solvable in S.
I thought of strengthening H i n d m a n '
strengthened
(i) do not m e e t
Split the real n u m b e r s
Is the following true:
in the c o m p l e m e n t
belong to the c o m p l e m e n t
x + y, x (
many
1 < ~ < co (A0zlf-hA~z) < ~ 0
-c
then there is an infinite s e q u e n c e
f(n) for infinitely m a n y
class?
so that the equation
that all the
n
{Xn)X n<
a r e in the s a m e
numbers
One would even
is a set of integers
s e c o n d possibility w o u l d be:
Is there a s e q u e n c e
~kak?
Divide the integers into c o n t i n u u m
< f(n), for infinitely m a n y
nontrivial
split the integers into
a I < a Z < ..., a n < f(n) so that at least
the set of all s u m s
the initial ordinal of the continuum,
a
f(n) so that if w e
s theorem
in the s a m e
Is the following true:
Is there an infinite s e q u e n c e
w a y as S z e m e r 6 d i
Let
A
a I < a Z < ...
be a
a n d an
?
58
integer
t
observed
so that all the integers
a. + a. + t a r e in the s a m e c l a s s ?
Straus
l
j
that the full s t r e n g t h of H i n d m a n ' s t h e o r e m d o e s not h o l d in this case.
Some
t i m e a g o I thought of the following fascinating possibility:
the i n t e g e r s into t w o classes.
Is it true that t h e r e a l w a y s
Divide
is a s e q u e n c e
al, a 2 ....
so that all the finite s u m s
same
~ ¢ .a. a n d all the finite p r o d u c t s
II a.
a r e in the
x I
i I
A t this m o m e n t
the p r o b l e m is open.
M o r e g e n e r a l l y o n e c a n ask:
class.
Is t h e r e a n infinite s e q u e n c e
formed
from
answer
is n o but no c o u n t e r e x a m p l e
The
the
a's
a I < a Z < ...
a r e in the s a m e
following m u c h
a I < a 2 < ...
weaker
so that all the m u l t i l i n e a r
class?
One
would
perhaps
expressions
guess
that the
is in sight.
c o n j e c t u r e is also open:
Is t h e r e a s e q u e n c e
so that all the s u m s
class?
Perhaps
Graham
proved
we
a. + a
and products
a a. a r e in the s a m e
i
j
1 j
s h o u l d also r e q u i r e that the a
a r e also in the s a m e class.
1
that if w e
four distinct n u m b e r s
Hindman
proved
divide the integers
x, y, x+y, x y
that if w e
divide the integers
four distinct n u m b e r s
the s a m e
S o far nothing is k n o w n
Answering
there always
(i=j p e r m i t t e d )
a r e in the s a m e
Hindman
just i n f o r m e d
that this will b e c o r r e c t e d
found a decomposition
sequence
exists.
h a s density
theory),
is a n infinite s e q u e n c e
0.
class - Z 5 Z is best possible.
Z < t < 990
all g r e a t e r than
in c a s e w e
a q u e s t i o n of E w i n g s ,
the J o u r n a l of c o m b i n a t o r i a l
into t w o c l a s s e s t h e r e a r e
all in the s a m e
there are always
class.
< Z5Z
Hindman
that if w e
into t w o classes,
1
x, y, x+y,
assume
proved
xy
then
all of
all the i n t e g e r s
>__3.
(will a p p e a r s o o n in
divide the integers into t w o c l a s s e s
x I < x Z < ...
so that all the s u m s
x
i
+ x.
J
class.
me
that t h e r e m a y
b y the t i m e this p a p e r
into t h r e e c l a s s e s
In fact H i n d r n a n
b e a g a p in his proof,
appears.
A[, A Z, A 3
observes
but I h o p e
O n the o t h e r h a n d h e
so that no s u c h infinite
that o n e of his s e q u e n c e s
say
A 1
In his e x a m p l e
(i)
Al(X ) =
~
1 <
c x I/2 .
ai~ A 1
a,<x
1--
It is not yet clear w h e t h e r
More
than I0 y e a r s
(and to w h a t
a g o P~. L.
extent)
Grahaln
(1) c a n b e i m p r o v e d .
and I conjectured
that if w e
the i n t e g e r s into t w o c l a s s e s t h e n
(2)
1 = 1Z !x i
'
Xl < x 2 < "'"
(finite s u m )
split
59
is a l w a y s solvable with the
x. all in the s a m e class. This should probably not
1
generalisations are possible.
be too difficult. Clearly m a n y
References
n,
Neil H i n d m a n , Finite s u m s with s e q u e n c e s within cells of a partition of
J. C o m b i n a t o r i a l T h e o r y Ser A 17(1974), i-ii, J. B a u m g a r t n e r , ibid. 384-386.
80
7.
Let
1 _< a I < ... < an
minimum
number
r
a' s.
of the
A new
extremal problem
be a s e q u e n c e
of integers.
of distinct integers w h i c h are the s u m
I conjectured that for every
r and
If true this s e e m s
difficult a n d s e e m s
extremely
E > 0
D e n o t e by
fr(n)
the
or product of exactly
if n > n0(E , r), fr(n) > n
r-E
to require n e w ideas
(unless of c o u r s e an obvious point is being overlooked).
Szemer6di
and I observed
that it follows f r o m
d e e p results of F r e i m a n
that
~=~fz(n)/n
but even the proof of fz(n) > n
i+¢
= ~,
seems
to p r e s e n t great difficulties.
Z
fz(n) >
n
-
-
(log
is certainly false.
Perhaps
n
D e n o t e by
F(n)
2
n) k
the true o r d e r of m a g n i t u d e
of fz(n)
is
exp (-c log n/log log n) .
the smallest integer so that there are at least
integers w h i c h are the s u m
or p r o d u c t of distinct
a.'s.
It s e e m s
F(n)
distinct
certain that
I
F(n) > n
k
for
but I h a v e not b e e n able to p r o v e this.
n > no(k)
By a remark
of V~. Straus it holds for
k=Z.
#
It is not h a r d to see that
of
F(n)
exp (log n) c ,
(i) holds for e v e r y
T h e following m o r e
be a g r a p h of n
integer
Perhaps
the true o r d e r of m a g n i t u d e
is
(i)
perhaps
F(n) I/n --1.
c.
general p r o b l e m
vertices and
k
x i,x i ~ x., 1 < i < j < n.
j
the t w o integers
-1
edges.
j
and
m i g h t be of interest.
T o e a c h v e r t e x of G
If x.
is joined to x.
~
J
_
x. + x.
c > 1
x.x..
1 3
T h u s w e associate
Let
G(n;k)
w e associate an
w e associate to the edge
Zk
integers to the g r a p h
61
G(n;k).
ing to
and
D e n o t e by
G(n;k).
A(G(n;k))
the smallest n u m b e r
if log
log kn -- Z
Perhaps
then
of distinct integers c o r r e s p o n d -
A(G(n;k)) > n Z-E
for every
n > nO.
conjecture
This conjecture if true is a far reaching extension of m y
Z-c
fz(n) > n
All these conjectures
c a n be extended to the c a s e w h e n
the
x.
G > 0
original
a r e real
I
or c o m p l e x
numbers
or e l e m e n t s
For a few weeks
of a vector space.
I thought that the following result m i g h t hold (here
a n d our g r a p h is regular of d e g r e e one).
integers.
the
Zn
T h e n there a r e at least
numbers
Let
I
i
al,...,an;
(or at least
{ a +b ,a.b.}, i = I,Z ..... n.
I
too optimistic.
n+l
A.
cn)
b I.... ,b n
llubin s h o w e d t h a t
numbers
amongst
the
among
I was much
I
T h e conjecture certainly fails for
can be real n u m b e r s
Zn
distinct n u m b e r s
c > I/Z
a n d if the
!
b's
be
n = Zm
then there d o n o t h a v e
{ai+bi, aibi}.
to be m o r e
It is a l m o s t
than
cn I/Z
certain that the s a m e
a.'s
and
i
distinct
holds if
the
a. a n d b. are restricted to be integers, but as far as I k n o w R u b i n did not
l
i
i+~
yet w o r k out the details. If w e a s s u m e
k > n
or p e r h a p s only k / n --0o one
p e r h a p s m i g h t get s o m e
results but I do not h a v e a n y plausible conjecture
so far.
62
Some
8.
unconventional
Is t h e r e a s e q u e n c e
aI < a Z <
problems
...
on primes
of integers
satisfying
A(x) =
~
1<
a.<x
1
log x
so that all sufficiently l a r g e i n t e g e r s a r e of the f o r m p + a. ?
If this i s
1
i m p o s s i b l e then p e r h a p s such a sequence exists for which the density of i n t e g e r s
not of the f o r m p + a. is O. Clearly many s i m i l a r questions can be asked for
i
o t h e r s e q u e n c e s then the p r i m e s but t h e r e a r e v e r y f e w results.
Ruzsa proved
that t h e r e is a s e q u e n c e
integer is of the f o r m
of i n t e g e r s
Zk + a .
a I < a Z < .... A(x) < c x / l o g x
so that e v e r y
Is it true that t h e r e is s u c h a s e q u e n c e
for e v e r y
1
c<
log Z +
E
The
prime
integers w h i c h
k-i~ple conjecture
d o not f o r m
t h e r e are infinitely m a n y
primes.
This problem
infinite s e q u e n c e
states:
a complete
integers
Let
aI <
set of r e s i d u e s
n
rood p
so that all the integers
is u n a t t a e k a b l e at present.
of integers.
... < a~
It w o u l d
Let
are prime.
Perhaps
it w o u l d
for a n y
p.
n
reasonable
are
be an
b e interesting to find a n e c e s s a r y
be more
Then
{n+ai} I < i < k
a I < a Z < ...
sufficient condition for the e x i s t e n c e of infinitely m a n y
n + a.
be a set of
and
so that all the integers
to p e r m i t
for e a c h
n
a
1
finite n u m b e r
of exceptions.
should be mentioned
here:
An
Are
so that all but a finite n u m b e r
number
of p r i m e s
unfortunately
old a n d v e r y fascinating c o n j e c t u r e
there two sequences
of the s u m s
a
a r e of the f o r m
nobody
can prove
aI < a Z <
of O s t m a n
... ; b I < b Z < ...
+ b . a r e p r i m e s a n d all but a finite
z
3
T h e a n s w e r is o b v i o u s l y No'. but
a. + b ?
i
3
this. H o r n f e c k
showed
that both s e q u e n c e s
must
b e infinite.
Let us now
r e t u r n to o u r p r o b l e m .
If n
is s u c h that all the integers
n + a., i = 1,2 .... a r e p r i m e s w e first of all clearly m u s t h a v e
(n+a.) ~ (n+a.)
i
1
j
a n d w h a t is m o r e
(n+ai, n+aj) = i. Is it p o s s i b l e to find a n e c e s s a r y a n d sufficient
condition for the following t h r e e p r o p e r t i e s
of a n infinite s e q u e n c e
is a n o t h e r infinite s e q u e n c e
(ai+bj) ~ (ar+bs) ,
B
so that
i.
3.
a. + b. a r e all p r i m e s ?
I think p r o p e r t y
i
3
but I h a d n o t i m e to think this o v e r carefully.
reasonable
conjecture.
The
problems
may
1 and
For
change
There
(a.+b.,a + b ) = i,
I 3 r
s
can probably be handled,
Z
3
A?
Z.
the only h o p e w o u l d
if w e
permit
for e a c h
be a
a.
1
b. a finite n u m b e r
of e x c e p t i o n s also w e c o u l d restrict o u r s e l v e s
9
assume
(a.+b. , a.+b. ) = i for i <_jl < Jz < o0.
1
De
(l)
J1
Bruijn,
z
JZ
Turin
and I considered
f(n)
=
z
,.i
the function
. . . . .
p < n n-p
to asking:
and
63
It is not difficult to s h o w
(z)
that
!
x
It follows
from
z
f(n)-l,
± ~
x
n< x
fZ(n)-l.
n< x
n
l r ( n + n ~) - lr(n) > c 1--oog n
Hoheisel'sclassical
that
l i m inf f(n) > O.
i"i=0o
It is likely that
(3)
lira inf f(n) = 1 ,
n=o0
Perhaps
f(n) = o(log log n).
inaccessible
there
states:
is a
y < x
A weaker
To every
¢ > 0
conjecture
there
is an
which
x0
is perhaps
so that for
not quite
every
x > x0
so that
(4)
~(x) - ~(y) <
One
lira sup f(n) = o0
n=o0
in fact feels that
~ ~(x-y).
~r(x) - it(y) should be usually of the o r d e r
of m a g n i t u d e
x-y
a n d therefore it is r e a s o n a b l e to g u e s s that (4) is satisfied for e v e r y
log x
y < x - (log x) C for sufficiently large C. In fact I c a n not at this m o m e n t
disprove:
x-y
w(x) - ~(y) < c I log x
(5)
(5) w o u l d
imply
f(n) < c logloglog
n
could try to study
f(p)
y < x-
(log) C
and perhaps
li---~m ( n ) / l o g l o g l o g
We
for
n > O.
but this is e v e n h a r d e r
than
prove
1
~(x)
I conjectured
1
~i p-pj
(6)
where
o n c e optimistically
in
~i Pj < p " log p.
fZ(p) -- 1 .
p< x
that
- 1 + o(i)
(6), if true, is of c o u r s e
hopeless.
f(n).
I could not
64
H e n s l e y and P~ichards r e c e n t l y s h o w e d that if the p r i m e k - t u p l e
i s t r u e (in f a c t i t c e r t a i n l y " m u s t "
there are infinitely m a n y
absolute constant
x
for w h i c h
y
~(x+y) > ~(x) + it(y), and in fact for an
c > 0.
(7)
w(x) + w(y) + c Y / ( l o g y) < w(x+y) .
l~ichards a n d I h a v e a f o r t h c o m i n g
Monatshefte
der Mathematik.
There
p a p e r on s o m e
of these questions in
is an i m p o r t a n t d i s a g r e e m e n t
l~ichards believes that (7) holds for arbitrarily large values of
x
conjecture
of c o u r s e b e t r u e ) t h e n f o r e v e r y l a r g e
and
y.
c
b e t w e e n us.
a n d suitable
I conjecture the opposite.
O n e final conjecture:
consecutive p r i m e s
in
Let
n < ql < "'" < qk <--m
be the s e q u e n c e
of
(n,m)
k
1
<
i=l qi -n
for a certain absolute constant
c.
~
p<m-n
1
--+c
p
Trivially the opposite inequality is not true,
since there a r e arbitrarily large gaps b e t w e e n the p r i m e s .
It does not s e e m
to
be trivial to p r o v e that
(8)
lira inf ( ~ m
n=o0
z
n< qi < m
I
- -
qi-n
-
A)
x
p< m
= _~
p
A t p r e s e n t I do not see h o w to p r o v e (8).
Eggleton,
problems
Selfridge a n d I a r e writing a long p a p e r on s o m e w h a t
in n u m b e r
of our p r o b l e m s
theory.
(9)
We
conjecture
g(n) -- oo as
a k < k Z÷s
for
Let
(n-ak, n-at) = 1 for all
g(n) =
only p r o v e
(i0)
O u r p a p e r will a p p e a r in Utilitas M a t e m a t i c a .
related to (I) states as follows:
smallest integer for w h i c h
unconventional
n - - oo.
a 0 = 0, a I = I, a k
0 < i < k.
is the
Put
l
~ a.
i=l I
This is p r o b a b l y v e r y difficult.
k > (log n) C, C = C(~),
a k < C k log k if k > (log k)
but p e r h a p s
~C
We
One
can
65
where
~C
depends on
C.
Perhaps
(I0) is a little too optimistic,
" m u s t " ( ? ) hold if k > exp(log k) I/2
Straus and I conjectured:
primes.
T h e n for k > k 0
but (i0) certainly
w h i c h w o u l d easily i m p l y (9).
Let
Pl < PZ < " °"
there always is an
i < k0
be the s e q u e n c e of consecutive
so that
2
(ii)
Pk < Pk+iPk-i
Selfridge with w h o m
"
w e discussed this p r o b l e m
strongly doubted that (ii)
is true, in fact he e x p r e s s e d the opposite conjecture.
D e n o t e by
f(k) the n u m b e r
of changes of signs of the s e q u e n c e
Z
Pk - Pk+iPk-i "
Perhaps
f(k) --o0 as
k
0 < i< k .
tends to infinity, this of course w o u l d be a v e r y considerable
strengthening of our conjecture with Straus.
I cannot even prove
Z
A n old result of Tur~[n and m y s e l f states that Pk - Pk+iPk-i
li~n=sup f(k) = o0 .
has infinitely m a n y
changes of signs.
Put
%
= Pk+l - Pk"
both have infinitely m a n y
T u r g n and I p r o v e d that dk+ 1 > d k
solutions.
We
and
dk+l < ~k
of course cannot prove that d k = d~+ 1
has infinitely m a n y
solutions.
W e further could not p r o v e that dk+ Z > dk+ 1 > d k
has infinitely m a n y
solutions.
It is particularly annoying that w e could not p r o v e
that there is n___oo k 0
(IZ)
so that for every
i > 0.
d!<O+ i > dko+i+l if i - O(rnod Z) and
Perhaps
problems
dko+i < % 0 + i + i
w e overlooked a simple idea.
on consecutive p r i m e s :
if i ---l(mod 2)I '.
T u r i n has s o m e
Is it true that for every
d
very challenging
and infinitely m a n y
n Pn ---Pn+l ( m ° d d)?
Finally, in connection of our conjecture with Straus and Selfridge's doubts,
the following question of Selfridge and m y s e l f m i g h t be of interest:
be a sequence of positive density.
l<i<
(13)
Is it true that for infinitely m a n y
k
Z
a k > ak+iak_ i ?
Let
k
a I < a Z < ...
and every
66
D o e s (13) hold if the density of a's is i?
References
I. Ruzsa,
O n a p r o b l e m of P. ErdBs,
Canad.
Math.
Bull. 15(1972), 309-310.
Ira. ErdSs and P. Tur{n, O n s o m e n e w questions on the distribution of p r i m e
n u m b e r s , Bull. A m e r . Math. Soc. 59(1948), ZTI-Z78, see also P. Erd}Js, O n the
difference of consecutive primes, ibid 885-889.
P. ErdSs and A. R4nyi, S o m e
S i m o n Stevin 27(1950), 115-126.
P. ErdSs and K. Prachar,
Univ. H a m b u r g 26(1962), 51-56.
p r o b l e m s and results on consecutive primes,
S~tze und P r o b l e m e
~ber
Pk/k,
Abh. Math.
Sea.
P. ErdSs, S o m e applications of graph theory to n u m b e r theory, Proc.
second Chapel Hill conference on c o m b math. , North Carolina, Chapel Hill, N C
1970, 136-145.
P. Erd$s, S o m e
(1972), 91-95.
p r o b l e m s on consecutive p r i m e n u m b e r s ,
D. Hensley and Ian Richards,
(1974), 375-391.
Primes
Mathenuatika 19
in intervals, Acta Arithmetica 25
67
9.
Many
Some
extremal problems
extremal problems
explain w h a t I h a v e in m i n d
< a
< n
k(n)-
distinct.
in real a n d c o m p l e x n u m b e r s
on integers can be extended to real n u m b e r s .
consider the following p r o b l e m :
be a s e q u e n c e of integers.
Then
P r o b a b l y there is a
(Z)
c
maxk
n) 3/Z
Assume
that the products
a a. are all
i j
< max
k (n) < w(n) + c I n 3 / /
og n) 3/Z
so that
(n)
=
w(n) +
but (Z) will not c o n c e r n us now.
real n u m b e r s .
1 <__a I < . . .
(0 < c Z < Cl)
w(n) + c Z n 3 //4(/l o g
(i)
Let
To
Assume
c
n3/~
Let
/n 3/4/
)
og n) 3/Z + o \
/ (log n) 3/Z
I <__a I < . .. < ak(n) <__n
be a s e q u e n c e of
that
lauav - atasl >l
for every choice of the indices
prove
u,v,t,p.
Does
(i) r e m a i n true?
I cannot even
k (n) = o(n).
Clearly nearly all the e x t r e m a l p r o b l e m s
during m y
long life can be extended in this way.
munbers
or m o r e
lhil < n
be
n
generally m e m b e r s
complex
numbers
in n u m b e r
In fact the a's
of a vector s p a c e
assume
theory w h i c h I c o n s i d e r e d
e.g.
could be c o m p l e x
Let
h I ... h k,
that
lhahb - hchdl ~ 1
holds for e v e r y
1 5 _ a , b , c , d <_k.
but at this m o m e n t
How
large can be
If the h's
I do not see it.
max
k ?
o(n Z) is really certain
are c o m p l e x integers a result like
(Z) can undoubtedly be proved.
Now
I discuss s o m e
more
such questions.
A s e q u e n c e of integers
a I < a Z < ...
is called a primitive s e q u e n c e if a i ~ a.. Primitive s e q u e n c e s h a v e
3
b e e n investigated a great deal see e.g° our s u r v e y p a p e r with S~rk~zi a n d S z e m e r e d l .
I
But p r o b l e m
problem
an
e > 0
(1) is not yet solved for primitive sequences.
(I) one could characterize the s e q u e n c e s
and a primitive s e q u e n c e
A s a first step to solve
n I < n Z < . ..
a I < . . . for w h i c h
for w h i c h there is
•
68
A ( Z nk) > E Z n k
for every
k = l,Z,...
T h e generalisations
problems:
every
Let
to real s e q u e n c e s
a I < a Z < ...
seem
be a s e q u e n c e
to lead to interesting diophantine
of real n u m b e r s
and assume
that for
i,j,k
(3)
Ika i - ajl >__ 1.
I cannot e v e n p r o v e that (3) implies
A(x)
lira - -
A(x) = E l)
= 0,
x
a<x
1
O n e w o u l d guess that m o s t
sequences
of the a s y m p t o t i c
properties w h i c h a r e valid for primitive
also hold if only (3) is a s s u m e d .
T h e only result is the following unpublished t h e o r e m
that the
fact is not true for primitive s e q u e n c e s
believe that m u c h
assumed
of J. Haight.
a's are rationally independent a n d satisfy (3). T h e n
of integers.
A(x)/--
/
of the difficulty will already be e n c o u n t e r e d
if the
lecture at Q u e e n s
College one m e m b e r
a's a r e
of the audience
S. Shapiro) a s k e d the following question w h i c h I h a d overlooked:
be a s e q u e n c e
of real n u m b e r s .
Assume
J ]l ii
i
Is it then
true
Let
(perhaps
1 < aI < ...
that
II aj lZl
j
for e v e r y pair of distinct choices of the finitely m a n y
13 ..
J
x
In v i e w of Haight'sresult I
to be rational n u m b e r s .
During my
and
Assume
--0. This in
non-negative
integers
~.
i
that
(1)
~: 1 =A(x)<_~(x)
?
a .<x
1--
(I) is certainly a fascinating conjecture.
Beurling p r i m e
numbers
The
a's
are sometimes
a n d h a v e a large literature - as far as I k n o w
n e v e r b e e n c o n s i d e r e d before.
A v e r y nice a n d unpublished
called
(i) has
conjecture of Beurling
69
states: A s s u m e
that the n u m b e r
is x + o(log x).
T h e n the a's
l h a v e to apologize if m y
not always r e m e m b e r
of n u m b e r s
satisfying
not exceeding
x
references are s o m e t i m e s inaccurate but one does
things
O n c e the following beautiful p r o b l e m w a s
D o e s there exist an infinite sequence of distinct G a u s s i a n p r i m e s
I did not r e m e m b e r
T h e conjecture w a s told m e
theory m e e t i n g 1963 N o v e m b e r
Motzkin.
i
w h o suggested a p r o b l e m in a discussion w h e r e m a n y
[Yn+l " Y n I < C ?
cleared this up.
IIa.
i
are the primes.
are m e n t i o n e d in rapid succession.
attributed to m e :
of the f o r m
w h o told m e
this but E. Straus
by M o t z k i n at the P a s a d e n a n u m b e r
and it w a s apparently raised by Basil G o r d o n and
I naturally liked it very m u c h
and told it right a w a y to m a n y people,
naturally attributing it to Motzkin, but this w a s later forgotten.
T h u s the p r o b l e m
is returned to its rightful owners.
T h e following p r o b l e m w a s considered by G r a h a m
S
be a m e a s u r a b l e set in the circle
two points of S
is an integer.
How
S~rkBzi and myself:
Let
IYl < r and a s s u m e that no distance b e t w e e n
large can be the m e a s u r e
of S ?
S~rkSzi
has the sharpest results, but nothing has been published yet.
References
P. ErdSs, O n s o m e applications of graph theory to n u m b e r theoretic
problems, Publ. l%amanujan Inst. i(1969), 131-136, see also S o m e applications of
graph theory to n u m b e r theory, T h e m a n y facets of graph theory Proc. Conf.
W e s t e r n M i c h i g a n Univ. K a l a m a z o o 1968 Springer Verlag, Berlin 1969, 77-8Z.
P. Erd}Js, A. S~rkSzi and E. Szemer4di, O n divisibility properties of
sequences of integers N u m b e r theory Colloquium Bdlyai Math. Soc. N o r t h Holland
Z(1968), 36-49.
70
I0.
Let
the number
Some
more
unconventional
problems
a. < a. < ... b e a n infinite s e q u e n c e
1
I
of solutions of
of integers.
Denote
by
f(n)
V
n
=
~
a.
Is there
a sequence
rightly
for which
criticized
seems
very
as
being
strange
and
f(n) -- 0o as
artificial
attractive
and
.
1
i=u
n--
0o.
This
problem
in the backwater
to me.
If
a. = i
can
perhaps
of Mathematics
then
f(n)
be
but it
is the number
of odd
1
divisors
of
n.
probabilistic
Iknow
of no
methods
Leo
example
but they
Moser
and
where
do not
I considered
f(n) >__ Z
seem
the
for
all
n > nO.
I tried
to work.
case
where
the
a.
are
primes.
We
con-
1
j e c t u r e d that
We
[ i m f(n) = 0o
d o not e v e n k n o w
MacMahon
(1975), 9 2 2 - 9 2 3 )
sequence
a n d that the density
that the u p p e r
and Andrews
consider
x.'s
(i. e.
density of the integers w i t h
(see G.
E. A n d r e w s ,
the following p r o b l e m .
of integers w h e r e
consecutive
xn
U
x
Iknowit
I could
Let
xI < x Z <
is not
Andrews
= (1+o(1))
even
not settle
...
be
known
f(n) > 0
Math
(lower
that no
density)
is positive.
Monthly
32
b e the
is not the s u m
of
that
n log n
loglog n
x
/
- all I could
x
is the
Jn
sum
-- o0 .
do is to ask
a few
of consecutive
other
x.'s.
n
that the density
exists.
1 = x I < x 2 < ...
conjectures
whether
this question
such
Let
f(n) = k
1
n
far as
Amer.
is the s m a l l e s t integer w h i c h
x n ~ ~x.).
1
As
of integers w i t h
questions.
Is it true
1
of the
x.'s
is
O?
I am
not
sure
about
the density
1
but would
be very
surprised
if the
lower
density
would
not be
O.
v
Assume
the density
that
x
now
of this
that all the
sequence
> c n log n must
is
hold
sums
O.
Ex
are
u i
It is obvious
for infinitely
distinct.
I am
by a simple
many
n,
thus
now
averaging
the lower
confident
that
process
density
is
O.
n
It is not hard
to show
that for these
sequences
1
E
Z
n<x
--<
x .1
C
<n
i
]
holds
for
to prove
an absolute
or disprove
constant
this.
C.
Perhaps
Z --=-~
X.
1
converges
but I do not see how
71
Let
x I = 1 < x Z < ...
v
~x.
sums
u
I am
Xn
is the s m a l l e s t integer for w h i c h
a r e distinct.
A
simple
counting a r g u m e n t
gives
x
finite a n d infinite
I am
have time
cn
for all
n.
B Z
3 -- 0. All the p r o b l e m s
on
n
c a n b e a s k e d h e r e too, but a l m o s t nothing is
sequences
not really s u r e h o w
more
question:
I < ... < a k ~ n
consecutive
a's.
Hofstadter
of i n t e g e r s
then
consecutive
sequence,
m,
n<
m~
L(n) <
L(n)
integer so that for e v e r y
which
is not the s u m
s e v e r a l of his p r o b l e m s
sample
as follows:
What
a I = 1, a Z = Z.
is the a s y m p t o t i c
of his p r o b l e m s :
Let
of the s e q u e n c e ,
behavior
a I = Z, a 2 = 3.
subtract
Does
were
inspired
Define a
If a I < ... < a n
is the s u m
of t w o
of this s e q u e n c e ?
Form
1 and append
l~epeat this o p e r a t i o n indefinitely.
which
of his p r o b l e m s :
a n + l is the s m a l l e s t i n t e g e r w h i c h
a's.
of
C n?
is a s m a l l
a I < a Z < ...
a r e a l r e a d y defined,
distinct e l e m e n t s
b e the s m a l l e s t
told m e
Here
Xn/
carefully.
L(n)
Is it true that
b y this q u e s t i o n of U l a m .
Another
Let
t h e r e is a n
l~ecently D.
with
difficult t h e s e q u e s t i o n s a r e since I did not
so far to investigate t h e m
One
or more
<
n
s u r e that t h e r e is s u c h a s e q u e n c e
sequence
all the
3
1
known.
l~a
where
all p r o d u c t s
of t w o
these elements
to the
this s e q u e n c e
h a v e positive
density?
A f e w d a y s ago,
a sequence
1 ~ a I < a 2 < ...
a 2 ..... a k
number
Kenneth
are already
a
a I = i, a n d the n u m b e r
Let
+ a. < x.
1
x
x).
Then
is a l w a y s
put
> x.
1 < i < j < k
of solutions of
hoped
a
+ a
which
would
Unfortunately,
a. + a. < x is less t h a n
1
j-that this a n d a g o o d deal m o r e is true.
Let
is a n
l<u
al,...,a n
nO
<v<
that
the
(or a 0 = 0,
< x,
0 < i < j < k, j > 0
of solutions of
of solutions of
show
we
that m y
There
considered
theorem
J
--
with Fuchs
is
that the n u m b e r
is little doubt,
though,
the following question:
Is it true that t h e r e
v
s o that for n > n O the n u m b e r
of distinct s u m s of the f o r m
.~
l=u ai'
n, is less t h a n ~ n Z ?
W e p r o v e d this if a. =i, but c o u l d not attack
i
the g e n e r a l
case.
of the i n t e g e r s
a. + a. < x
i
j-< x will then
a. + a
c o u l d not e v e n p r o v e
x + o(x).
(of 1976) H a r h e i m
be a permutation
is less
i--
that the n u m b e r
that the n u m b e r
of solutions of
in S e p t e m b e r
x - k
1
x + o ( x I/4+~)
essentially best possible.
Early
is less t h a n
/
b e of the f o r m
assume
Define
j--
x = a k + I. O b s e r v e
Rosen
--
a I = i,
b e the s m a l l e s t i n t e g e r for w h i c h
1
than
the following construction:
inductively as follows:
defined.
of solutions of
l~osen told m e
l,Z,...,n.
72
References
P. ErdBs and W. H. J. Fuchs, O n a p r o b l e m of additive n u m b e r
J. L o n d o n Math. So¢. 31(1956), 67-73.
theory,
EXPLICIT
FORMULAS
IN THE T H E O R Y OF A U T O M O R P H I C
FORMS
C. J. Moreno.
CONTENTS
Introduction
§ i.
§ 2.
§ 3.
. . . . . . . . . . . . . . . . . . . . . . .
Automorphic
Representations
and Euler P r o d u c t s
. . . . . . . . . .
i.i.
Classical
1.2.
Automorphic
1.3.
Relation
1.4.
Langlands'
1.5.
The F u n c t i o n a l
1.6.
Some E x a m p l e s . . . . . . . . . . . . . . . . . . . . . . . .
Rankin's
Automorphic
Forms
Between
75
81
Forms . . . . . . . . . . . . . . . . .
on Adele
Modular
81
Groups . . . . . . . . . . . . . .
Forms
and Forms on Adele
Groups
8Z
. .
Euler P r o d u c t s . . . . . . . . . . . . . . . . . .
Equation
Convolution
2.2.
The Constant
Term Matrix
Group
. . . . . . . . . . . . . . . . . . . . . . . .
2.3.
Some Euler P r o d u c t s
2.4.
The A v e r a g e
Zeros
in the Critical
3.1.
The H a d a m a r d
3.2.
The First v o n M a n g o l d t
3.3.
Explicit
and
for the E i s e n s t e i n
their F u n c t i o n a l
Size of the E i g e n v a l u e s
Series
Equations
of H e c k e
99
99
of the
109
.....
Operators
. . .
Strip . . . . . . . . . . . . . . . . . . . .
Product
Estimates
Formula
. . . . . . . . . . . . . . . .
Formula
9Z
95
Method . . . . . . . . . . . . . . . . . . . .
The I n g r e d i e n t s . . . . . . . . . . . . . . . . . . . . . . .
Fo(N)
86
of Euler P r o d u c t s . . . . . . . . . .
2.1.
85
. . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . .
119
131
148
148
15Z
158
74
§ 4.
§ 5.
§ 6.
Explicit
Formulas . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.
The Second
4.2.
Examples
von M a n g o l t
. . . . . . . . . . . . . . .
[70
Formulas . . . . . . . . . . . . . . . .
181
L(s,~) . . . . . . . . . . . . . . . . . . .
184
of Explicit
Zero Free Regions
for
5.1.
A Hadamard-Landau
5.2.
Prime N u m b e r
5.3.
The P r o b l e m
Type
Formula
170
Inequality . . . . . . . . . . . . . .
Theorems . . . . . . . . . . . . . . . . . . . .
of E x c e p t i o n a l
Zeros
. . . . . . . . . . . . . .
184
193
197
Zeta D i s t r i b u t i o n s
. . . . . . . . . . . . . . . . . . . . . . . .
199
6.1.
Introduction
. . . . . . . . . . . . . . . . . . . . . . . .
199
6.2.
The E x p l i c i t
Formula
Z0[
References
and W e y l
Symmetry
. . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ZI5
75
Introduction
The purpose of these notes is to study the analytic properties
products associated
to automorphic
adele ring of the rationals.
classic trilogy
[ 30 ].
representations
where
~
function
T(n)
does not vanish on the
Secondly it is shown that the Euler product asso-
ciated with
T(n) 2,
continuation
to the whole s-plane with a pole at
at first only defined for
Re(s) > ii,
s = ii.
about the average behavior of
has a meromorphic
Lastly it is shown
T(n) 2
gives some informa-
T(n).
In the years following the publication
of Rankin's work, only his third
paper received much attention becaus~ of its connection with Ramanujan's
ture
IT(p) l < 2p II/2.
at least for automorphic
representations
have received little attention up till now.
GL2~A),
The first paper seems to
It is our intention in these notes
to extend these results of Rankin to the Euler products
tations and to develop applications
that the full sig-
of the adele group
of Rankin's second paper seems to have been realized.
of automorphic
represen-
to a circle of ideas that have their origin
in the proof of the prime number theorem.
applications
conjec-
It is only in recent years, when harmonic analysis has
yielded a novel point of view to the theory of modular forms,
nificance,
is the
There it is first shown that the Euler product associated
line of absolute convergence.
tion about the size of
GL2~),
The genesis of this work is to be found in Rankin's
to the well known Ramanujan arithmetical
that good information
of
of Euler
The results we obtain have important
to the study of the arithmetical
properties
of automorphic
represen-
tations.
To set the stage for our subsequent discussion,
main results for the classical
we now explain Rankin's
theory of modular forms.
The Euler product associated with the function
~(s) = (2~)-SF(s)
~
P
T(n)
is
(i - T(p)p -s + pll-2s)-l,
76
where
T(n)
is the
n-th coefficient of the q-expansion
oo
q ~
co
(i - qn)24 =
n=l
In
[ 30 ]-I
[ T(n)qn.
n=l
Rankin proved, by a meLhod that is also applicable to the Euler
products of all holomorphic cusp forms that are eigenfunctions of the Hecke
operators,
that
~(s)
does not vanish on the line
Re(s) = --.~
In these notes
we show that indeed Rankin's method applies to the more general class of Euler
products associated to cuspidal automorphic representations
of
GL2~A); further-
more, we will obtain zero-free regions which are similar to those that arise in
the study of Riemann's zeta function
already exploited by Rankin in
~(s).
[ 30 ]-II,
A key idea in our >roof is the fact,
that the coefficients of the Dirichlet
series associated to the Euler products satisfy the Petersson-Ramanujan
a fact now established by Deligne
resentations of
GL2~A)
conjecture,
[ 7 ] only for those cuspidal automorphic rep-
over the rationals whose component at infinity belongs
to the discrete series, on the average.
In particular we prove a result, well
known to experts but not stated explicitly in the literature,
that implies that
the coefficients in the Fourier expansion of a real analytic cusp form satisfy
the Petersson-Ramanujan
conjecture on the average.
The proof of this result is
along the same ideas as in [ 30 ]-III, and the crucial point is the possibility
of obtaining sufficient information about the analytic nature of the Euler product associated with the Dirichlet series n~la(n)2n -s.
had done in [ 30 ]-II
when the
a(n)
This is indeed what Rankin
are the Fourier coefficients of a holo-
morphic cusp form which is an eigenfunction of the Hecke operators;
respect it must be proved that the Euler product for
real point on the line of absolute convergence,
from the point of view of hamonic analysis,
of the real analytic Eisenstein series at
a(n)
2
in this
has a pole at the
and this is seen, in retrospect
to depend on the distinguished pole
s = 1
and the fact that the Petersson
inner product of a cusp form with itself is essentially the residue of the
Dirichlet series of
a(n) 2
at the pole.
cuspidal automorphic representations
~
More precisely we prove that given two
and
~'
of
GL2~A) one can associate an
77
Euler product
L(s,g×7')
whose local factors are 3-dimensional,
be the Euler product of an automorphic
a meromorphic
continuation
representation
GL3(/A), [ ii ],
and has
to the whole s-plane and has no zeros and at most one
simple pole on the line of absolute convergence.
representation
of
that seems to
associated with
When
T(n), L(s,~ ×~)
~ = ~'
is the automorphic
is, after a trivial normalization,
oo
2 -s
n~iT(n) n .
the Euler product of
Other results that we prove are generalizations
mulas of number theory.
of the von Mangoldt
In particular we study the distribution
Euler products on the critical strip and also prove various
formulas like the following
for the Ramanjuan
function
1"'#(2)(0)
log
for-
of zeros of the
types of explicit
T(n):
(x-l)
X~
-
T(p a) log p = -~ ~(i)(0 )
0 -~-'
p <x
where the sum
E
P
is taken over the zeros of
~(s).
From this formula and the
results already mentioned we obtain the estimate
T(p) log p << xl3/2exp{-c(log
p<__x
which in a weaker form follows from Rankin
[ 30 ]-I
x)½},
and had been hinted at in the
Princeton version of Hardy's Twelve lectures on Ramanujan
tion of our explicit
elliptic curve;
formulas is to the Hasse-Weil
representation
log p = ~ ~ - +
p~<_x
N(p ~) = Card(EOFp~))
L(s,E)
L(s,E)
of an
is the Euler
~
- 2L(2)(0'E)
" - -
+ log (x-l),
L(1)(0,E)
runs over the non-trivial
zeros of
The interest in such a formula arises from the intriguing possibility
that the multiplicity
meaning
and the sum
Another applica-
then
XP
(p~ + 1 - N(p~))
L(s,E).
zeta function
indeed it follows from our results that if
product of an automorphic
where
[ 13 ].
[ 5 ].
of the zero at
O = 1
does have a definite geometric
In this direction we prove that, for the Euler product L(s,~)
of a cuspidal automorphic
representation
~,
the order of vanishing
of L(s,~)
78
at the real point on the critical line, which is the point of major interest in
the case of elliptic curves, is bounded from above by the logarithm of the conductor
(= level in the old terminology)
of
7.
Another problem that motivates our study of the analytic properties of
the Euler products of automorphic representations, which is not entirely unrelated
to the results we have stated above, is the possibility of carrying out for these
Euler products the program first initiated by Riemann for
die Anzahl der Primzahlen unter einer gegebenen Grosse"i
~(s)
in his "Ueber
It is to be hoped that
a closer study of Euler products from the point of view of harmonic analysis may
give some information concerning Riemann's observation that Euler products only
vanish on the symmetry line of the functional equation.
An initial contribution
that we make in this direction is the derivation of an explicit formula for the
Euler products of automorphic representations,
(automorphic)
representations
as Well
of extensions of
GL 1
[ 41 ] had done for the
of Galois type.
Our explicit
formula exhibits an element of Weyl-symmetry which closely resembles the constant
term of an Eisenstein series associated to a maximal parabolic subgroup of a semisimple connected Lie group.
The content of the various sections
the notation used throughout these notes.
is. as follows.
In
§2
In
§i
we introduce
we develop Rankin's convolution
trick [ 30 ]-II in a form which is most suitable to the applications we have in
mind.
§4
In
§3
we study the zeros of Euler products in the critical strip.
we obtain explicit formulas with error terms.
inside
the
critical strip are obtained.
In
Finally in
§5
§6
In
zero-free regions
we give a derivation of
our explicit formulas with Weyl-symmetry and give also a general formulation of
problems concerning the location of the zeros of Euler products of automorphic
representations.
One final remark about
representations
of the group
~.
In these notes we work mostly with automorphic
GL2~A) ,
where
~
is the adele ring of the rationals;
although most of our computations do not generalize to extensions of
~,
the
79
ideas do seem to apply, even to the case when
linear group over a number field.
GL 2
is replaced by a reductive
We are unable to carry out such an investi-
gation at the present time due to the absence of a concept of conductor of an
automorphic representation, general enough to include an explanation of the well
known shape of the local factors of Artin's Euler products at the ramified primes.
This we hope to pursue in future publications.
Another reason for working over
is the implications that Langlands' funtoriality principle [ 24 ]
has for the
explicit estimates we carry out in these notes and which in the setting of
GL 1
had already been hinted at by Lang [ 21].
Acknowledgment:
I am greatful to R. R. Rao, R. A. Rankin and J.-P. Serre who,
directly and indirectly, contributed many ideas developed here.
received from an NSF grant.
Support was
During most of the writing of these notes the author
was visiting member of the Institute for Advanced Study at Princeton and an associate member of the Center for Advanced Study at the University of Illinois.
would like to thank Nancy Lomax for her skillful typing.
C. J. Moreno
February 1977.
I
80
§i.
Automorphic Representations
i.i.
and Euler Products.
Classical Automorphic Forms.
We shall follow closely the notation of Gelbart's monograph
H
be the upper half plane, let
real
2 × 2
G = GL 2
and let
matrices with positive determinant"
~
= GL~)
define
all functions
g(z) = (az + b)/(cz + d)
and
f
define
on
H
and
g
in
~
]K
Let
be the group of
for a pair
g = (c
a b) E ~
'
z E H
[ i0 ].
d
and
1
j(g,z) = (cz + d)(det g)-2-. For
(fl [g]k)(z) = f(g(z))J(g, z)-k
for any integer
k.
For any positive integer
N
F0(N ) = {(a b) ~ SL2(2Z):
c _= rood" N}.
For any two positive integers
space
M~(N,g)
k
and
N
(MI)
f
is holomorphic on
(M2)
fI [~]k = ~(a)-if
f
on
E
and any character
of modular forms of weight
of all complex valued functions
let
k,
H
level
N
of
and type
and observe that by
g-IF (N)g
(M2)
consists
H
for all
F (N)
E
the
satisfying
~ = (~ ~) E F0(N).
The next condition deals with the regular behavior of
To state it we let
(~/N) x
be the kernel of the homomorphism of
fI[y] k = f
is commensurable with
for all
F0(N) ,
~ E F (N).
f
at the cusps.
r0(N):
I~ ~ ) ÷ g ( d )
Now for any gEGL2(Q) ,
and hence
g-iFe(N)g A PZZ = {(~ ml): m E 2g }
for some integer
n depending on
triangular unipotent matrices).
g
(where
Since
P
is the algebraic group of upper
g-iF(N)g
fixes
fl[g] k
and
f
is
81
holomorphic we have a Laurent expansion
2 gi mz
fl[glk(Z ) = ~ am(g)e
m
n
The regularity condition is then
(M3)
C
For every
g
GL2(Q)
A,
and positive real number
there exists a constant
such that
Ifl[g]k(x+iy)
for all
- a0(g) I j c exp(-2~y)
y > A.
The set
Sk(N,E)
of cusp forms is the subspace of functions in
%(N,E)
having the further property that
(M4)
s0(g) =
for all
1.2.
g E GL2(Q)
and
f
~/m
(fl[g]k)(z+nx)dx = 0
z E H.
Automorphic Forms on Adele Groups.
Let
~ = ~Q
denote the adele ring of the rationals and recall that the
idele group has the decomposition
/Ax
= Qx.~R+ ~
(2g )x;
P
also recall that the unique factorization of an idele and the strong approximation
law for
SL2(Q)
imply that
GL2(/A) = GL2(Q)'GL2@R) ~
where
Kp
is any choice of subgroups of
GL2(Q p)
Kp,
such that
Kp = GL2(2Zp)
for
82
all but a finite number of
p,
~x
P
p.
is surjective for all
In our case
{(: ~) E GL2(~p):
divide
N.
Kp
is open for the rest and the determinant map into
will be the closure of
c E 0 mod. N}.
It is
F0(N)
GL2(~ p)
GL2(~ p)
precisely when
Koo is taken to be the orthogonal group
thought of as the discrete subgroup of
in
G/A = GL2~A)
02(~).
p
or
does not
GQ = GL2(Q) is also
consisting of Q-rational
points.
The center
Z/A
(resp. ZQ)
of
GL2~A)
(resp. GL2(Q))
scalar matrices and therefore is isomorphic to the ideles
Let
p
denote the right regular representation
morphic form on
GL 2
is any function
~
on
~x
of
consists of
(resp. QX).
GL2~A).
An auto-
G/A satisfying the following
conditions.
(i)
(ii)
(iii)
¢(yg) = ~(g)
for all
for some grossencharacter
~
is right
lates
p(z)~,
GL~ (~),
z
G
for
k E K
= GL2(~)
i.e.
for all
z ~ Z/A;
the space of functions on
is finite dimensional;
alone,
~
is smooth and the trans-
in the center of the universal enveloping algebra of
C
and
for any compact set
M
Q
in
GL2~A) and
A > 0,
there exist con-
such that
0
for all
O(z)~ = ~(z)$
is finite dimensional;
(v)
stants
o(k)~
as a function of
for
@
K = K • ~Kp-finite,
G/A spanned by the translates
(iv)
y E GQ;
g e ~, a ~/A x
with
lal > A.
An automorphic
form
~
clal M
is called a cuspidal form if it also satisfies
83
the condition
f ~((~ ~)g)dx = 0
Q\~
for almost all
The space of cusp forms is denoted by
A0(~).
We also denote by
L2(G~Gt~,~)W/~ the Hilbert space of measurable functions
(i)
(ii)
~(yg) = ~(g)
for all
p(z)~ = ~(z)~
g.
~
on
G/A such that
y E GQ
for all
z C Z/A
and
(iii)
f
l~(g)12dg < ~.
Z/AGQ\G/A
The subspace of
by
L2(GQ\%, ~)
consisting of cuspidal functions is denoted
L20(GQ\G/A,~). It should be noticed that
subspace of
L2(GQ~G/A,~) consisting of
A0(~)
K-finite,~-finite functions
center of the universal enveloping algebra of
unitary representation of
G/A in
GL2~)).
L2(GQ~%,~)
GL2~A)
if it occurs in some
Let
pC(g)
(~is
the
denote the
given by right translation.
An irreducible unitary representation of
representation of
coincides with the dense
GL2(A)
p~.
is called an automorphic
Recall that we have a decom-
position
pC = f~Sds@(~.~J )
J
into a continuous part and a discrete part.
An automorphic representation is
called cuspidal if it is equivalent to a discrete component of
fact that any irreducible unitary representation of
GL2~A)
[ i0], p. 76); we write such a representation in the form
pC.
We use the
is factorizable (cf.
~ = ® ~ , where
P P
runs over all primes including the infinite one, and for each
p,
~
P
p
denotes an
84
irreducible unitary representation
of the local group
Gp = GL2(Qp) o
The repre-
sentations which are of interest to us are those which have almost all their local
components of class
1.3.
i.
Relation Between Modular Forms and Forms on Adele Groups.
When convenient we shall make use of the isomorphism between the complex
modular variety
GL2(Q)\GL2(/A)/~XK
and
F\H
g = g0googf ÷
where
and
K
F
is an open compact subgroup of
is the inverse image in
SL2(Q)
into
GL2(/Af) , /Af
SL2(Zg)
of
morphic forms on
K
~x
the ring of finite ideles
under the canonical injection of
is identified with the subgroup
b2
The map that takes holomorphic
the subset
z = goo(i),
GL2(/Af) (cf. [ 3 ], p. iii);
a2
given by
GL2(/A)
is given by
GL 2 OR) p ~ Kp
of
GL2~A )
cusp forms of
f ÷ ~f,
Sk(N,~)
on
H
to autog = goo'gc in
where for an element
we put
~f(goog c) = (fl[goo]k)(i)gA(g c);
here
gA
is the grossencharacter
following prescription:
~A = p ~ gp
canonical homomorphism from
putting
putting
(~ ~ ) ~
~p(a).
~Xp
to
The function
~f(%g) = ~f(g)
be an automorphic form on
H,
of
for any
GL2~A).
~x
and
determined by
Cp
(~ /N)X"
~f
~
according to the
is the pull back of
gA
is extended to
is extended to all of
y C GL2(Q).
g
The new function
by the
~
Kp
GL2~A)
~f
by
turns out to
If we start with a real analytic form
then by the same prescription we get an automorphic form by letting
~f(g~gc ) = (f[[g~]o)(i)gA(gc).
by
f
on
85
We shall make full use of the one-to-one correspondence between the
eigenfunctions of the Hecke operators on the space of holomorphic cusp forms
or real analytic cusp forms which are new forms and automorphic representations
(cf. [i0 ], p. 94, Theorem 5.19).
1.4.
Langlands' Euler Products.
We now review briefly Langlands' construction of Euler products from
automorphic representations (cf.[~3 ], §2). The basic details for this construction can be found in Satake [ 32 ].
concern
G = GL2,
the following construction
particular we may take
For
p
elements and
K
Although most of the applications we make
G
to be a Chevalley group.
a finite prime let
Gp = G(Qp)
the maximal compact subgroup
P
p = ~
we put
works for more general groups; in
G
= G~R),
be the group of
G(~ ).
P
the group of real points and
Qp-rational
For the infinite prime
Koo the maximal compact
subgroup of G~, say corresponding to the involution associated to a Chevalley basis.
As usual the adele group
for all primes
p
G/A is the restricted direct product of the groups
with respect to the compact subgroups
subgroup of principal ideles in
primes
p
is clearly a compact subgroup of
K = ~ K
p
P
GQ
is the discrete
taken over all
G/A. Let L2(GQ\~A)
be the space of
all square integrable functions on
GQ\~A
lations by elements of
Let ~
be the semisimple Lie algebra of
Let
be the Caftan subgroup of
K.
a Cartan subalgebra o f ~ .
~
G/A. The product
Kp.
.
Fix a Borel subgroup
consists of the
B
containing
~ • L2(G~G/A )
/
T
which are invariant under right trans-
T.
G
with
G
and +
Lie algebra
The subspace of cusp forms L~(G~G/A)
with
~(ng)dn = 0
Gp
for all
g E G/A,
86
where
N
for all
is the unipotent radical of the parabolic
P
except
G.
For a prime
subgroup
P
containing
p, which may be infinite,
H
B,
will denote
P
the algebra of all compactly
supported regular Borel measures on
G
which are
P
invariant under left and right translations
by elements
Kp;
(cf.
multiplication
is given by convolution
define the operator
%(~)
on
L~(GQ\G
u
/A)
%(~)~(g)
of the compact subgroup
[ 9 ], p. 278).
If
~ E Hp,
by
= f ~(gh)d~(h).
G
P
If
~
%(f)
p
is the measure associated
instead of
%(D)
all the measures
in
to a function
and consider
H
f
f E LI(Gp)
as an element of
are absolutely
we sometimes write
H .
P
For a finite prime
continuous with respect to Haar measure.
P
2
L0(G~G/A)
The space
is, for all
~i
p,
admits an orthonormal
an eigenfunction
generate an automorphic
%(~)
representation
We consider an element
morphic representation
of
~
basis
~i,~ 2 .....
for all
of
~ E Hp;
to it.
the translates
~ = ~
For a measure
~i
of
i
~ .
/~GI^ which we denote by
of this basis and let
that corresponds
such that each
be the auto-
~ E H
we let
P
~(~)~ = ~ ( ~ ) ~
and observe that the map
~ ~ Xp(D)
gives a homomorphism
of
Hp
into the complex numbers.
Let us now recall how all such homomorphisms
into the complex numbers arise.
Borel subgroup
B
Observe that, since
T /T A K
P P P
containing
N \B
PP
Let
T ~d
N
be the unipotent
~
to
T ,
P
determines
and
any homomorphism
a homomorphism
B
w:
%
can> N \B --+ T /T n K
w-w-+~.
P
P P
P P
P
P
of
Tp = T ( Q p ) .
w
of
B
into the
P
complex numbers which we again denote by
H
radical of the fixed
N p = N(Qp) ' Bp = B(Qp)
put
is isomorphic
into the complex n u ~ e r s
of the Hecke algebra
87
If
of
ad.b
b
belongs
to ~ ,
to
B
let
D(b)
the Lie algebra of
be the
N.
determinant
Since
G
P
can be written as a product
bk
of an element
= B K ,
P P
b E B
of the restriction
any element
g E G
P
and an element
P
k E K .
P
Set
~w(g) = w(b)ID(b)[½.
The function
~
is well defined and any other function
w
(l.1)
on
G
satisfying
p
~(bgk) = w(b) iD(b)i½p(g)
for all
b,g
satisfying
e H
~
and
(I.i)
define
P
k
is a scalar multiple
are parametrized
~(~)~w
of
~w;
by elements
in fact all the functions
w ~ Hom(Tp/TpNKp,
~).
For
by
(%(~)Pw)(g)
= / ~w(g h)dD(h)"
G
P
The function
scalar
%(~)~w
Xw(~).
satisfies
The map
and all homomorphisms
(i.i)
~ ÷ Xw(~)
of
H
and so
%(~)~w = Xw(~)~w"
then defines a homomorphism
which are continuous
for some
of
H
P
to
~,
in the weak topology are
P
obtained in this way.
a
u
The homomorphism
in the Weyl group so that
Suppose
;
p
is finite.
there is a homomorphism
Xw
equals
w(t) = w'(t q)
Let
from
L
Xw,
for all
if and only if there is
t E T .
P
be the lattice generated by the roots of
T /T A
P P
K
or
P
from
T
to
CL = Hom(L,2Z)
P
so that
] ~ ( t ) I = p%(t)(~)
if
~
is a root.
Here
~
is the character of
T
associated
to
~.
If
~
is
88
a root let
~
be the coroot attached to
~.
Let ~l,~2,...,~n be the simple roots
and
(Aij) = ~(~i,~i)j
be the Cartan matrix o f ~ .
The matrix
(~i,~.)
(aij) = ( ~ )
is the transpose of
The lattice
CL'
(Aij) and is the Cartan matrix of another Lie algebra
generated by the roots of a split Cartan subalgebra
can be identified with the lattice i n ~
in such a way that the roots of
Also
CL = Hom(L,~)
generated by the eoroots
correspond to the elements
can be regarded as a lattice i n + ~ .
can in fact be regarded as the lattice of weights of
Similarly,
~]R
may be identified with
the lattice of weights o ~ .
algebra
c~
and let
an isomorphism
CT
o ÷ c
Let
CG
Hom(CLJR)
c~
of
el,~2,...,~n.
so ~
eL'
so ~ I R D L' D L,
be the Cartan subgroup corresponding to
T
in
G
and
D CL D CL'.
if
L'
be the simply connected group with
of the Weyl groups of
c~
~l,~2,...,~n
It contains
~
c~.
~.
is
Lie
There is
with that of
CT
in
CG
such that
cu(%(t)) = %(ot),
If
w E Hom(Tp/TpAKp,
w(t) = ~%(g)
CT
for all
associated to
%.
~),
t.
t E T .
P
then there is a unique point
Here
% = %(t)
and
~%
g E CT~
is the rational character of
Thus associated to each homomorphism of
complex numbers is an orbit of the Weyl group in
so that
CT;
H
into the
P
or equivalently we may say
that to each such homomorphism there corresponds a semisimple conjugacy class in
the complex group
CG~.
Let us now consider an automorphic representation
~ = ~
P
of
G/A in
89
2
\
L0(G Q G/A) which is unramified everywhere, i.e. each local representation
a class one representation.
To an automorphic form
~
corresponds, for each prime
p,
of
finite let
{gp}
a homomorphism
be the conjugacy class in
Xp
CG~
in the space of
Hp
into
~.
corresponding to
be a finite dimensional complex representation of
CG~
~p
~,
If
Xp.
is
there
p
is
Let
r
and consider the Euler
product
~(s,~,r) = ~ det(l - p-Sr(gp))-l,
P
the product being taken over all finite primes.
that this product is absolutely convergent for
shall see later on, in the particular case
Langlands
Re(s)
has shown ([23 ], §3)
sufficiently large; as we
G = GL 2, Re(s) > 1
To the prime at infinity one also associates a
is enough.
F-factor.
Let
l
be the
homomorphism
T /T N K
which is such that
in
~,
I~(t)I
= e l(t)(a)
every homomorphism of
Hom(L,]R)
+~IR =
if
T /T N K
~
is a root.
into
~
Since
L
is a lattice
is of the form
w(t) = e l(t)(X)
for some
X E ~.
Thus to every homomorphism of
an orbit of the Weyl group in
~
Hoo
into
~
there is associated
or a semisimple conjugacy class in
is the homomorphism associated to the automorphic form
corresponding conjugacy class and let
dim. r
det(l - r(X)T) = ~
(i - li(°°)T)
i=l
9,
let
{X}
c~.
If
be the
90
be the characteristic
product
~(s,Z,r)
polynomial of
r(X).
F-factor
that goes with the
is
F(s,~,r)
dim.r s-l.1
s-~.
= ~
~
2
F(~).
i=l
The Euler product associated
the finite dimensional
to the automorphic
complex representation
L(s,~,r)
It is expected,
The
r
of
representation
CG E
and
is
= r(s,~,r)~(s,~,r).
and known in many cases, that
L(s,~,r)
satisfies
a functional
equation of the type
L(s,~,r)
where
E(~,r)
gredient of
is a complex number of absolute value
r.
In some known instances,
automorphic
representation
exponential
factors that depend on
The delicate
~,
G = GL2,
the dual group
of the group
[ 2 ].
r
is the contra-
is allowed in the
may contain exponential
CG
CG~ = GL2(~).
is given in Langlands'
it suffices
factors
duals are the
In our particular
An excellent
[24 ],
to remind the reader that
their corresponding
A,C,B,D,E,F,G.
construction with many interesting variations
Report
and
s.
A,B,C,D,E,F,G
types
1
ramification
g(~,r)
for our purposes
Chevalley groups of types
complex groups of respective
where
the number
construction
p. 25, in great generality;
for
%
= g(~,r)L(l-s,~,r),
introduction
case of
to Langland's
can be found in Borel's Bourbaki
gl
1.5.
The Functional Equation of Euler Products.
In the following we consider only automorphic representations of
Let
~
be such a representation and for a finite p~ime
its local components.
The conductor
f(~ )
P
of
~
p,
let
~
P
GL2~A).
be one of
is defined by the following
P
theorem of Casselman ([ 4 ], p. 302):
Theorem.
Let
~
be an irreducible admissible infinite dimensional
P
representation of
ideal
f(~p)
of
GL2(Q p)
~p
with central character
4.
such that the space of vectors
Then there is a largest
v
with
~p((~ bd))V = ~(a)v
for all
(a
c b)
d C F0(f(~p) ) = {(a bd) E GL2(ZEp):
is not empty.
c - 0 mod f(~ p )} '
Furthermore, this space has dimension one.
We will say that a local representation
The global conductor
f(~)
~
P
is ramified if
of an automorphic representation
f(~) = ~
~
f(~ ) # ~ •
P
P
is defined by
pordpf(~p),
P
where the product runs over the ramified primes.
The construction of the Euler products associated to automorphic
representations of
GL2~A)
can be done in various ways (cf. [ 2 ],[ i0 ]).
Here
we follow a combination of the method presented in Gelbart ([ i0 ], p. 113)
with
the method of Langlands
described in
§1.4.
First we consider the unramified
92
situation.
If
p
is a finite prime and the local representation
~
P
belongs to
the principal series then it is parametrized by two quasi-characters of
x
Qp: ~l(X) = Ixl sl, ~2(x) = IxlS2; if
space ~(~I'~2)
-
~0
is any
K -invariant function in the
P
of all locally constant functions @ on G
such that
P
q~((~l t2) g ) = ~l(tl)~2(t2) I
for all
coset
]½#~(g)
is the characteristic function of the double
tl't2 @ QX and if T
P
P
Kp(P 01)Kp' Kp = GL2(2Zp)
then the convolutions
@0*Tp(g) = / ~0(xy-I)Tp (y) dy
G
P
= p½(pSl + pS2)@0(g).
To such a local representation we associate the conjugacy class
{gp}
in
GL2(~)
which contains the matrix
~p = (~sl ~s 2)
and to a finite dimensional complex representation
r
of
GL2(~)
we associate
the local factor
Lp(S,~p,r) = det(l - p-Sr(~p))-l.
To this local factor there corresponds a trivial root number
the quasi-characters
~i
and
~2
g(~ ,r) = i.
P
are both ramified then we put
Lp(S,~p,r) = 1
and the root number is taken to be, when
r = r2
the standard 2-dimensional
If
93
representation of
GL2(~) ,
g(~p,r 2) = W(~I)W(~2),
where
~i"
W(~i)
is the root number of the local Tate zeta function associated to
If only one of the
~i' say D2'
is ramified, we take for local factor, when
r = r2,
1
Lp(S,~p,r 2)
l-~l(p)p -s
and the root number is taken to be
to the special representation and
be
1
and
C(~p,r2) = W(~2).
~i
~(~p,r 2) = W(~I)W(~2).
is ramified then
Otherwise, if
=
Lp(S'~p'r2)
and
g(~p,r 2) = W(~2).
representation
~
If
p
= ~(~i,~2)
L (s, ~ ,r 2) =
where
%'i = -r.1 - m.l if
g(~ ,r 2) = i 2.
If
~
If
~i
~p = ~p(~l,~2)
Lp(S,~p,r 2)
belongs
is taken to
is ramified, we put
1
l_Dl(p)p-S
is the infinite prime then for a principal series
we put
-½(S-%l) F S-%l) -½(s-X2) s-%
(~
r(~J~)
~i(x) = Ixl ri sgn(x) mi.
= ~(~i,~2 )
For the root number we take
is a discrete series representation then
L (s,Z ,r2) = ~ - ½ ( S - % l ) r ( ~ l ) v - ½ ( s - % 2 ) r ( ~ - ~ )
where
%1 = -Sl
number we take
Let
and
%2 = -Sl - 1
if
~i(x) = Ixl si sgn(x) ni.
For the root
~(~ ,r2) = iSl-S2 +I.
S
representation
are unramified.
be the special set of finite primes
~
p
for which the local
= p(~l,~2) is a special series representation and DI,~2
P
We define the special conductor of ~ and the special root
number, respectively, by
f0(~) = ~ p ,
p~S
g0(~,r2 ) = (-1) ISl]-~l(p) ,
pES
94
where the second product is taken over all the quasi-characters
in the special representations
~
= p(~l,~2)
for
~i
that appear
p E S.
P
The global root number associated to an automorphic representation
of
GL2~A)
and the standard
2-dimensional representation
r2
of
GL2(~) is
given by
g(]~,r2) = ]-~ g(~p,r2).
P
The Euler product associated to
~
and
r2
is
L(s,~,r2) = -~- Lp(S,~p,r2).
P
By Jacquet-Langlands
([ 16 ], p. 350, Theorem ii.i) we know that if
cuspidal automorphic representation of
the Euler product
L(s,~,r2)
GL2~A)
~
is a
with central character
4,
then
represents an entire function, is bounded on vertical
strips of finite width and satisfies the functional equation
L(s,~,r2) = g0(~,r2)f0(~)l-sg(~,r2)f(~)½-SL(s,~,~2 ),
where
1.6.
r2
is the twisted contragredient representation
~-ir.
Some Examples.
The Euler Products of Hecke.
a Dirichlet character of
(~/N) x
a holomorphic cusp form of weight
k
Let
k
and
and assume
on the group
N
be positive integers and
(-i) k = 4(-1).
F0(N).
Suppose
eigenfunction for the Hecke operator
fiTp =
oo
co
~ a qn + ~(p)pk-i ~ anqpn '
n=l ~
n=l
Let
p ~ N
f(z)
f
is an
be
95
and of the operator
U
P
oo
=
fIUp
~ a
qn
n= 1 pn
with the corresponding eigenvalues being
,PIN,
a . Define the zeta function of
P
f
by
co
[ a n -s
n
¢(s,f) =
n=l
= ~
To the cusp form
f(~) = N
f
(I-app-S)-I p~N (l-ap p-s+~(p)pk-I-2s)-I"
corresponds an automorphic representation
~f
of conductor
whose Euler product is none other than
S-%l
~s-X2
s-%2
k-l,
s-X~ _ ~i_Xi~
L(s,~f,r2) = ~ - ( ~ ) F ( - - 2 ~)~
~F(~)~(s+-f),
where
k-I
XI
2 '
k+l
%2 = -
2
i
The functional equation is
L(s,~f,r2) = e(~f,r2)N½-SL(s,~f,r2) ,
where
~A
Ig(~f,r2) i = I.
Incidentally, when
A
is the Ramanujan modular form and
is its associated automorphic representation then
g(~A,r2) = i,
f(~A ) = i,
and
L(S,~A,r2) = 2 ( 2 ~ ) - ( s + ~ ) F ( s + ~ )
~ T ( n ) n - S - ~ -.
n=l
This is an example of an Euler product associated to an automorphic representation
96
~A =
®P7 P
component
which is unramified everywhere.
~
Also in this example the infinity
is a member of the holomorphic discrete series.
The Euler Products of Maass.
field of discriminant
d.
Let
CK
the two element Galois group of
Let
K = Q(~)
be a real quadratic number
be theidele class group of
K/Q.
Let
E
K
X((~)) =
To each rational prime
p
and
G = {I,T}
be a fundamental unit of
the real Dirichlet character associated with the extension
an unramified grossencharacter of
K
K/Q.
Let
K
and
X
be
whose value at a principal ideal is
4
~ik/l°g
we attach a conjugacy class
{gp}
in
GL2(~)
with
det(l 2 - Tr2(gp)) = 1 - ap T + @(p)T 2
where the coefficients
a
are defined by
P
ap = k(~) + X ( ~ )
if
(p)=~.~T
ap = 0
if
(p) = ~ .
To the infinite prime we associate the eonjugacy class
X
~ GL2(~)
whose char-
acteristic polynomial is
. k~
.2
2
det(l 2 - Tr2(X ) ) = i + (l--~--~g
g) T .
The resulting Euler product
s-~ l
s-~ z
L-S'~K'r2-() = ~- - - 2 - - r ( ~ ) ~ - ~ F ( s - a 2 ) 2
where
~ det(l-p-Sr2(gP))-l'
P
97
~ik
log E
%1
and
%2
~ik
log
satisfies the functional equation
L(s,~K,r2)
= E(~K,r2)d
½-s
L(l-S,~K,r2).
This is the Artin-Hecke L-function associated with the 2-dimensional
of the Weil group
WK/Q
obtained by inducing the character
Recall that the Weil group
WK/Q
of the pair
K, Q
X
from
representation
CK
to
WK/Q.
is the group extension
i -+ C K ÷ WK/Q ÷ G ÷ i
obtained from the distinguished
L(S,~K,r 2)
o K = @pOp
H2(G,CK ) .
is in fact the Euler product of an automorphic representation
whose infinity component is a principal series representation.
automorphic form associated to
Tne map
generator of the cyclic group
~K ÷ OK
§2
[ 26 ].
in number theory and has been analyzed
we will consider other examples of
of
Maass
([ 16 ]).
with automorphic representations
representations
is one of those considered by
is of great significance
in depth by Langlands
In
oK
The
GL2(~)
of
GL2~A)
other than
r 2.
Euler products associated
and finite dimensional complex
g8
§2.
2.1.
Rankin's Convolution Method.
The Ingredients.
Let
N
be a positive integer and let
gruence subgroup.
with
s
Fix an eigenvalue
pure imaginary or purely real between
character of
of functions
~/QX
of conductor dividing
~
2
Lo(GQ/G/A,g) ,
in
Casimir operator
A
with
K 0 = p]~< K P
g
P
-i
N.
G = GL2,
for all
and
the natural
and
g
W (N,%)
we have
K P = {I~ bd) E GL2(~p):
g
Let
be a grossenthe subspace
such that under the action of the
g E GL2~A), r(@) E K
restriction of
i.
Denote by
at the 'infinity' component
~(gr(@)k 0) = C(ko)~(g)
where
be the usual Hecke conl-s 2
of the Casimir operator; assume % = 4
%
Fo(N)
to
and also
= S02(I~) and
c ~ 0
K .
P
A~ = %~
mod. N}
W (N,%)
k
and
E K0,
g = ~
Cp,
has the structure of
a finite dimensional Hilbert space with the inner product
(~i,~2) =
/
~i'~2
dg.
Z/AGQ\ G/A
The natural isomorphism
Z/AGQ\ ~A/K K0 $ F 0 (N)\ SL 2 (~)/S02 OR)
gives a correspondence between functions on the group and functions on the upper
half plane:
~(g) ÷ f(z)
adele group element
g
with
z = g~(i),
where
at the infinite prime.
g~
is the component of the
Under this correspondence the
above inner product is the relative Petersson inner product
(f(z),g(z)) =
/
f(z)g(z)d~,
D0(N)
is the
SL2-invariant measure on the upper half plane and
where
d~ = y-2dxdy
D O (N)
is a fundamental domain for
F0(N).
Hecke operators
T
P
and
T
P
acting
99
on the space
W (N,~)
are defined as usual ([ i0 ], p. 88 for the adele setting
and [26 ], §4. for the classical case).
W (N,%)
and
generated by functions
dINN~; let
W+
Let
W- (N,~)
s
g(dz), where
g(z)
be the subspace of
is an element of
be the orthogonal complement of
Ws(N,%)
in
We(N',%)
WE(N,%).
In the following the elements of Wc(N,%) will be viewed as functions on
a b
the upper half plane. Let A
( v d~ ' 1 < v < D0(N) = N p ~ N (i +--i) run over
v
Cv v
P
a representative system of elements in F (i) which correspond to a complete set
o
a
of inequivalent rational cusps
let
be t h e s m a l l e s t
KV
Fo(N).
v = O
v
rational
for the group
v
number
K
A simple calculation shows that
1-r 2
= 4
,
cusp
c
p
For each such
Kv(C~,N) = N.
Now if
g
in
o
v
1 K)A-I E
Av(0 1 v
for which the matrix
then the Fourier expansion of a function
z = x + iy
W (N,~)
and
about the
has the form
gl[p]o(Z) = ~
½
ap(n)y Kr(
K
is the modified Bessel function.
r
expansion of a function
cients
a(n)
g(z)
) exp .--~---~,
P
If we want to consider the Fourier
only about the cusp at infinity, there the coeffi-
will be written without any subscript except possibly to denote
their dependence on the function
definition:
~2~inx~
2 ~K
P
n#0
where
Fo(N).
a new form
g(z).
f @ Ws(N,%)
For convenience we introduce the following
is a non-zero element in
a common eigenfunction of all the Hecke operators
T
with
W~(N,%)
which is
(p,N) = I;
the
P
function
f(z) E Ws(N,% )
the cusp at infinity has
Remark 2.1.1.
is said to be normalized if its Fourier expansion about
a(I) = i.
As was already pointed out in
~i,
a new form
in the above sense corresponds to an automorphic representation
GL2~A Q)
whose local component
~
~ = ~
fEWs(N,% )
P
of
at the infinite prime belongs to the principal
series and whose restriction to the maximal compact subgroup
02~R)
is trivial.
Rankin's convolution method, which we explain below, can be applied also to
automorphic representations where the restriction of
~
to the maximal compact
100
subgroup is not trivial; we will not consider here this case in order to avoid
complications of notation that result from having to introduce a Bessel function
whose structure is more complex than that of the modified
Remark 2.1.2.
If
f E Ws(N,% )
is a normalized new form whose Fourier
expansion about the cusp at infinity has coefficients
Dirichlet series
~(s,f) =
~ a(n)n -s
n=l
Ks(Z).
{a(n): n C ~ },
then the
has the Euler product expansion
~(s,f) = q ~ N (i- a(q)q-S) -I p ~ N (l-a(p)p -s+£(p)p-2s)-l.
As in
§1.4,
if we put
L(s,f) = ~-½(S-%l)F(S-%l)~-½(s-%2)F(S;~2)~(s,f),
2
with
l-s
%1 = - r +
and let
then
~
(-i__~)
2
%2 = r +
'
be the automorphic representation of
L(s,f) = L(s,~)
is the Euler product
2-dimensional representation of
GL2(~) ,
I-E (-i)
2
GL2~A)
L(s,~,r2),
associated with
with
r2
f,
the standard
and it satisfies the functional equation
L(s,Z) = £(z)N½-SL(I-s,~),
where
E(~)
is a constant of absolute value
representation
If
k
1
and
~
is the contragredient
~(g) = ~(g)-I (g) ([i0 ], p. 116).
is a positive integer and
%
k(k-l)
2
,
denote the space of holomorphic cusp forms of type
properties of new forms in
HE(N,%)
we also let
{Fo(N),k,E}.
H (N,%)
The concept and
which we shall use in the following are
developed at great length in Winnie Li's article [25 ].
Here we recall the well
101
known fact ([ i0 ], p. 91)
morphic representation
that a new form in
~ = ~p
of
H (N,k)
GL2~AQ)
corresponds
to an auto-
whose component at infinity belongs
to the holomorphic discrete series.
Another important ingredient that is used in Rankin's convolution method
is the theory of Eisenstein series for
Kubota's book [ 17 ]
Let
rp
be one of the
SL2(~)
stabilizer in
Yo(N)
o = (ac b)d
j(o,z) = cz + d.
Im(z) = y.
Let
If
k
go(N)
rational cusps of
which caries the cusp
of the cusp
in
SL2~R )
rp, i.e.
and
z = x + iy
z
i~
into
Fo(N)
rp.
and let
Let
Fp
p
be
denote the
Fp = {4 E Fo(N): o(rp) = rp}.
For an
a complex number, we write as usual
is a point in the upper half plane we put
be a positive integer and
We extend
X
Po(N);
denotes a complex variable.
s
The basic reference here is
from which we borrow freely the following results.
an element of
element
~ (N).
o
to a character of
Fo(N)
X
a character defined modulo
by putting
X(o) = X(d)
To the data
{s,N,x,k}
for
N.
o = (ac bd) E
we associate the
Eisenstein series ([17 ], p. 63)
Ep(z,s,x,k ) =
~
X(O)~(p-lO'z)
OEF hE
P
i)k(imp-lo(z)) s,
"lJ(p-lo'z)
where the sum runs over a complete set of coset representatives
modulo
r .
P
r%
has the form
~p,XyS
6p,%
r = r (N)
o
We recall that the constant term in the Fourier expansion of
about the cusp
where
of
+
~p, ~(s,X)kyl-S
is the Kronecker delta function,
kl
(-i) 2Z2F (s) F (s-1)
~p, %(s'X)k =
k
k
r (s+~) r (s-y)
~p, %(s,X),
E
P
102
and
~p,Tt( s, X) =
X(Pd%-l) le1-2s,
~
(* *
O= c d )
where
• d
* )
(c
runs over a complete
which are inequivalent
plete set of inequivalent
If
cusps for
¢(s,x)
that this square matrix,
the analytic
in
Foo = { (0 1 ): n 6 2g }
modulo the group
by right and left multiplication.
Observe
set of coset representatives
continuation
p
and
F (N)
o
under its action
run independently
then the constant
over a com-
term m a t r i x is
= (¢p,X(s,X)k).
w h i c h plays an important
of the Eisenstein
Here w e shall use the following
~
p-lro(N)k
of
series
role in the theory of
Ep(Z,S,X,k) ,
theorem whose proof is identical
has
~o(N)
to that in
rows.
§6.2
in Kubota's book:
T h e o r e m 2.1.
If the rows in the column vector of Eisenstein
series
~(z,s,x,k ) = t(El(Z,S,X,k ) ..... E~o(Z,S,X,k)
have the same order as the rows in the constant
functional
term matrix
~(s,x) ,
then the
equation
~(z,s,x,k ) = ~(s,X)~(z,l-s,x,k)
holds.
Remark 2.1.3.
argument
similar
the constant
If
X
is the principal
to that given by Kubota
character and
([17 ], p. 45)
term m a t r i x has a simple pole at
s = i.
k = O,
then an
shows that each term in
In all other situations
103
Ep(Z,s,x,k)
s = i.
is regular at
Ep(Z,S,X,k)
not so then the residue of
independent of
z
To see this we simply observe that if it were
and also
at
s = 1
would be at the same time
X-automorphic and this is impossible.
After these preparations we are now ready to look at a typical example of
Rankin's convolution method.
Theorem 2.2.
Let
~
and
~'
be automorphic representations of
associated respectively to a holomorphic new form
11
k(k-l)2'
=
Wg2(N2,12)
modulo
k
i- r 2
2
12
induced by
about each cusp
in
HgI(NI,I I)
an integer, and to a real analytic new form
with
N
f(z)
r
P
gl~2 .
of
Let
N = ~.c.m. (NI,N 2)
and let
Suppose the Fourier expansions of
F (N)
o
g(z)
X
GL2GA Q)
with
in
be a character
f(z)
and
g(z)
are given by
2~inz
fl[p]k(Z ) = n~lap(n) e
<0
and
2~inx
gl[p]o(Z) =
Put, for
~ bp(n)y½Kr(2~K--~0) e
n#0
Re(s) sufficiently large,
Lp(S,~X~')
s+~
= ~p.'
-½rKP~[~]
and let ]L(s,ITxI[')
~(s,x)
F(s+~+r)F(s+~-r)
F(s + ~)
k
~
i
~ a (n)b (n)n s->~(k-l)
n= 1 @
P
be the column vector
t(Lpl(s,~
Let
KO .
x
~') ..... Lp (s,~
x
Z')).
be the constant term matrix for the Eisenstein series
Then the vector function
~(s,~ x ~')
Ep(Z,S,X,k).
has a meromorphic continuation to the
whole s-plane and satisfies the functional equation
104
• (s,~ × ~') = ~(s,×)~(l-s,~
where
~
and
~'
are the contragredients
Remark 2.1.4.
then the functional
If
rX
x
~
and
runs through the
equation for
Lp(s,TT
of
L(s,~
~T')
x ~')
x ~'),
~'
~o(N)
respectively.
rational cusps of
F (N)
o
can be written also in the form
= ~ q~p,l(s,x)L%(l-s,~
x
~'),
r%
where
~p,%(s,x)
is the
Remark 2.1.5.
the group
F (N)
o
p-%
entry in the constant
in the functional
equation for the function
S-matrix
~(s,X)
times a relatively
~(s,~
x 7')
cusp
rp
of
L (s,~ × 7')
p
seems to be only of a notational
In fact there is in the literature ample evidence
that the vector valued function
s
~(s,x).
The apparent scattering about each rational
which is brought about by the
nature.
term matrix
([ 8 ],[ii ],[14 ],[25])
is really a scalar valued function of
simple vector valued function.
The scalar function is most
likely related to theEuler
product constructed in §i.~. for anautomorphic representa-
tion of
Below we shall give explicit examples of such Euler products
GL2~AQ) x GL2~AQ).
satisfying relatively
simple functional
In the development
have, as a notational
of Rankin's
device,
equations.
convolution method it is convenient
the formal identity between fundamental
to
domains
which is given by the following lemma.
Lemma 2.3.
Let
be a rational cusp of
the stabilizer
of
p
D (N)
o
F (N)
o
be
F
p
be a fixed fundamental
and let
p E SL2(~ )
= {o E F (N): O(r) = r}
o
domain for
be such that
and put
F (N).
o
p(i ~) = r.
Let
Let
r
105
where
H
K
O
is a positive integer.
Define a region
S
P
in the upper half plane
by
K
Ixl < ~ ,
Sp = {z = x + iy:
y _> 0},
th en
Sp E
[
P-IODo (N)
o E ro(N)/r p
up to a set of
d~-measure zero, where the sum runs over a complete set of coset
representatives
of
Y (N)
o
Remark 2.1.6.
be applied to
modulo
The above congruence identity between regions, which will
F -automorphic
P
functions,
as an excercise for the reader.
Lemma 2.4.
have for
Re(s)
F .
p
is relatively easy to prove and is left
(See [ 30 ], p. 367).
With the same assumptions and notation as in Theorem 2.2. we
sufficiently
large
k
/ y2fl [p]k(z)g---r[P]o(Z)ySd~
S
K
= K 7TI(TZ~
~]
p
4#
where
k-i
2
P
S+--
£(s+½(k-l)+r)F(s+½(k-D+r)
£ (s +k)
co
~ ap(n)~ p (n)n_S_½(k_l)
n= 1
d~ = y-2dxdy.
Proof.
We multiply the Fourier expansion of
conjugate Fourier expansion of
gl[p]o(Z)
fl[p]k(Z)
by the complex
and integrate the product with respect
106
to
K K
in the interval "
~[--~'~]'
zz
x
where
K = Kp,
to obtain
K
S
fJ[P]k(z)g~[P]o(Z)dx
K
2
2~ny
oo
= K ~ a (n)bp(n)y½Kr(~)e
n= 1 O
where
y~+
z = x + iy,
2
S
y > 0.
K
We now multiply both sides of the above equality by
and integrate the resulting expression over the interval
respect to
y.
We evaluate explicitly
[0, ~]
with
the Bessel integral by using the well
known identity ([ 27 ], p. 92)
S0e-aXx~-iK
valid for
Re(~+~) > 0
(ax) dx = ~½(2a) -~ F(D+~)F(~-~)
r(~+½)
and a real; thus we get
o~k+s_2
K
SY
S~-
fJ [P]k(z)g~[P]o(Z)dxdy
0
2
n=l~ap(n)bp(n)n
F(s+k)
"47
= Lp(S,~ x ~').
The interchange in the order of summation and integration is justified
for
Re(s) > Oo
by the fact that
gl[p]o(Z)
and
fJ[p]k(Z)
are
0(y c) (resp.
--C v
0(y
))
constants
uniformly in
c
and
c'.
x
as
y ÷ ~
(resp. as
y ÷ 0)
with suitable positive
107
Proof of Theorem 2.2.
By lemmas (2.3)
and
(2.4)
we have
k
Lp(S,~×~') = f y2fl[P]k(z)g-TT~o(z)ySd~
S
P
k
=
~
OEFo(N)/~
~
=
f i
Y2fI[P]k(z)g-TT-PTo(z)ySH~
p- ~Do(N )
/
k
y2fl[P]k(z)g-TT~o(z)ySd~o(p-lo).
a~ro(N)l ~ Do(N)
We now use the transformation formulas
f ~-~-$-~J
raz + b~ = ~l(d)(cz + d)kf(z)
and
raz + b~
g~c-~-~-~j = E2(d)g(z),
which hold for any
~ E F (N),
to obtain
o
k
(Imz)2f][p]k(z)g~[p]o(Imz)Sd~o(p-lo)
= (Imz)~f(z)g(z)gl(~)~2(o)l j(p-lo'z) l)k(imp-lo(z))Sd~,
~lj(p-lo,z)
where
j(T,z) = cz + d
if
T = (c d)-
This change of variable applied to the
last integral gives
k
Lo(s,~X~') = /
y2f(z)g(z) Ep(z,s,x,k)d~ ,
D
(N)
o
where we have put
108
Ep(Z, s,x,k )
X(o) lj (p-lo,z)] k (Imp -I o(z) )s.
~
=
oE £o(N)/rp
Now the functional
<rj(p-lo,
@l~
equation for the Eisenstein
series
Ep(z,s,×,k)
can be written
as
Ep(Z,S,×,k) = ~ Cp,x(s,×)kEl(z,l-s,~,k),
where
~
runs over a complete set of inequivalent
this into the integral representation
of
cusps of
Lp(S,%X~')
We substitute
F (N).
o
to obtain
Lp(S,%X%') = Z ~p,%(s,X)kLl(l-s,#X#'),
where
~'
~
and
are the automorphic
representations
which are paired by the correspondence
now denotes
and
~'
if
Sl + ~ i
the vector function whose components
~(s, X)
is the constant
contragredient
and
g2 ÷ ~2"
are the functions
If
to
~
and
l(s,~X~')
L (s,~X~')
term matrix then we have
]L(s,~TX~')
= C>(s,x)]L(1-s,~TX~'),
which is what we wanted to prove.
2.2.
The Constant Term Matrix for the Eisenstein
It is possible to go further than Theorem
about the structure of the constant
Series of the Group
Fo(N).
(2.2) by using information
term matrix for the Hecke groups
£o(N).
The
results that we need are already available in the literature and are due to
Orihara
[ 28 ], §3.
is more convenient
Let
N
We now proceed to describe these results using a notation that
for our purposes.
be a positive integer and let
FN
denote
the principal
congruence
109
nl...
subgroup
of level
N.
Let
N = p]
n1
p]
be the faetorization
of
N
into
n.
distinct
primes.
We put
N = NiP i i,
1 < i <
i, and choose once and for all a
n.
set of integers
{d I .... ,d%}
such that
d. ~ 0 mod. Ni,
1
d. ~ 1 mod. pi l, ] < i <
1
%
The mapping
~
÷ ~
g i v e n by
X
{a I ..... a%} ÷ a =
~ d.a.
i 1
i=l
A
induces
a ring isomorphism
of
2Z/N
onto
ni
T7 z ~ / ( P i )"
i=l
Under
this mapping
we
have
(a(1),a(2),N)
= i
if and only if
(a (I)
(2)
i 'ai
,wi) = i
and
(u,N) = 1
Let
I = llX...xl%,
We identify
V(1)
I
if and only if
where
(ui,Pi)
I i = {(al,a 2)_ mod.
with a subset of
be the space of functions
(ZZ/N)X(TZ/N)
on
I.
= i,
1 < i < %.
ni
Pi : a l = i or a 2 = i, a I - 0 mod.
via the above
isomorphism.
Let
We then have
V(I) = V(II)@...@V(I%).
For each character
X mod.
N,
there exists characters
)<i mod.
n.
Pi I
such that
t
X(a) = ~ xi(ai).
i=l
Let
r i = r(xi)
+ 0 if
i < i < ~
there exists a primitive character
and r.l = 0
P
r.
X mod. " ~ p i 1
i=l
X(a) = x(a)
if
if
such t h a t
(a,N) = i.
D+I < i < ~.
Then
pi }.
110
If
T.
is a linear transformation
on
V(I.)
i
for
i < i < %
then
T = TI@...®T %
I
induces a linear transformation
on
V(1).
al
s = a--
associate the rational cusp
of
To each element
Fo(N),
s = {al,a 2}
in
I
we
and let
2
aI *
p = (a2 ,) ~ SL2(2Z)
be a representative
the stabilizer
of
al
p(i~) = - - .
a2
element with
s
in
F (N)
It can easily be verified
that
is
O
2
aln
l-ala2n
Fp = {(-a~n
.. N
l+ala2n):
n ~ ~ • (a~,N)};
clearly
F
is the stabilizer of
is
(a~,N).
correspond
p
s
nF
in
=
N
{(l-ala2n
-a~n
FN
2
aln
) n E ~ "N}
l+ala2n :
We now consider a distinguished
to each element
Ep(Z,s,x,k ) =
[Fp: Fp A F N]
and the index
s E I
set
of
the Eisenstein
functions
put in place of
If
p
on the element
the symbol
r = {a,b}
V(I)
in
Fp
by letting
~
X(O) fj(p-lO'z) )k(imp-ld(z))½(s+l)
de ro(N)/rp
~lj(£1d,z)I
imaginary axis the axis of symmetry.
Ep(z,s,x,k)
in
F P ~ FN
series
Notice should be made of the natural change of variable
series
of
and
s+l
s + T
which makes the
To indicate the dependence of the Eisenstein
{al,a 2}
of
I
we will find it convenient
'*'
r' = {a',b'}
are elements
a
<r,r')
= det.
a T
(b b ')"
in
I
we put
to
111
The main result of Orihara ([ 28 ], p. 141) can now be stated in a slightly generalized form as follows:
Theorem 2.5.
series
Ep(z,s,x,k )
With the notations as above we have that the Eisenstein
satisfies the functional equation
[F,,F, n rN]E,(z,s,x,k )
k
--
(_i) 2
s+l.
r (-~--)
A<s,~)
A(s+I,~)
-.s-k+l.
r (.s+k+l
~)
~ ~--T--)
9(i)®'" .~(%~F,,F ' n
FN]E*(z'-s'x'k)
where
^(s,X) : ~-½(s+A)r(S--~l)~ (l-x(p)p-S) -I
P
with
A = 1
if
X(-I) = -i
linear transformation of
~(i)(a,b)
and
V(I i)
% = 0
if
X(-I),
and for
i ~ D,
~(i)
is the
with matrix entries defined by
: -- ((a,b>, -sk-ni+k
Xi ~ ) P i
if
ki
p II a-b, 0 < k < n.-r.
1 1
Pi
0
and for
i > ~
otherwise
the linear transformation
s
~(i)(a,b ) = (X(Pi)Pi)
l-n i
~(i)
has matrix entries
Pi - 1
• ~( ) s+l
X Pi Pi
- 1
if
a
=
b
k-n.+l ~(Pi)P~ - l
= Pi
Remark 2.2.1.
Kubota ([ 17 ],P. 69)
l
~, ) s+l
" (~(pi)p~) -k
ktPi Pi - i
if
P~II a-b, O<k< ni-l.
The proof of this result is identical to that given by
for a similar result.
To make use of the ideas of Orihara
112
([28 ], p. 141),
one simply observes
that for a suitable class of functions
defined on the upper half plane and a Dirichlet
one has the simple combinatorial
aeFo(N)/rp
where
X(°)fo(z) =
X(o) = x(d) if
the constant
[Fp
Fp
:
1
[
l" n
P
d = (, d ).
and (2.5).
N
[
treated by
term matrix for the Eisenstein
Let the assumptions
Then the Euler product
to realize that
s+l
E(Z,T
,x,{al,a2} )
.
by the principle expounded
fu~(Z),
OeFN/F p n FN
It is then not very difficult
in Kubota
We can now state a more precise form of Theorem
Theorem 2.6.
defined modulo
×(u)
PeFo(N)/F N
FN]
term matrix for the Eisensten
n FN]Ep(Z , s,x,k)
X
identity
Orihara is related to the constant
[rp:
character
fo(z)
series
[ 17 ], p. 69.
(2.4).
and notation be as in Theorems
Lp(S,~×~')
satisfies
(2.4)
the functional
equation
l+s
[F,: F, n FN]L,(--~--,~×~')
k
(-l)2F(S--~l) 2
_.s+k+l.~.s-k+l.
Proof.
A(s,~)
}(1)®...@}G)[F,: r, n FN]L,(~,{ x~')
A(s+l,b
From the proof of Theorem
(2.4)
we recall the integral represen-
ration
k
.l+s
Lp[T,~x~')
=
f
y2f(z)g(z)Ep(Z,S,7,k)dQ.
Do(N)
Observe again that we have made a natural change of complex variable
the definition of the Eisenstein
series.
s +
s+l
We now multiply both sides of this
in
113
equality by the factor
the functional
[Fp: Fp n FN] ,
equation for
replace the symbol
[F,: F, N FN]E,(z,s,x,k )
O
by
'*'
and apply
given in Theorem
(2.5).
This proves the Theorem.
Remark 2.2.2.
s = 1
if and only if
under consideration
We have already remarked
X
is the principal
Lp(S,~×~')
that
Ep(Z,S,X,k )
character and
has no pole at
s = i,
k = 0.
has a pole at
In the case
= @~
since
and
P
7' = 8~'
P
have non-equivalent
components
if in the notation of Theorem
at the infinite prime.
More generally,
(2.4),
y½(l+s) + Cp(S,X)y½(l-s)
is the constant term of
Ep(z,s,x,k)
about the cusp
residues on both sides of the integral representation
p,
then a comparison of
for
.l+s
L[-~--, 7 x ~')
gives
that
.l+s
Residue L £ T , ~ × ~ ' )
s=l
where
< f,g)
associated
to
2 automorphic
Euler product
is the Petersson
~
and
7'.
inner product of the automorphic
forms
f
This identity which relates the equivalence
representations
L(s,~ × ~')
= (f,g) Residue ~p(S,X),
s=l
7
at
and
s = 1
~'
and
g
of the
with the analytic behavior of the
will play a fundamental
role in the investi-
gations to follow.
Theorems
and
~'
(2.4)
and
which are allowed
consequently
(2.6)
dealt with automorphic
to have a finite number of ramified local components;
the functional
contained rather complicated
equation satisfied by the Euler product
factors corresponding
simple functional
equation.
Lp(S,~ x~')
to the ramified components.
In contrast with this one can prove that in the unramified
satisfies a relatively
representations
situation
L(s,7×7')
114
Theorem 2.7.
Let
associated respectively
and
g(z)
in
~
and
~'
be automorphic representations
to real analytic wave forms
WI(I, %') ,
%' = l-r'
2 '
i.e.
f(z) in
WI(I'%)'
of
GL2(/A)
l-r
% = T
real analytic forms of level
i
which are eigen functions of the Hecke operators.
Suppose the Fourier expansions
of
at infinity are given by
f(z)
and
g(z)
about the cusp of
F (i) = F
o
f(z) = n~0~ a(n)y ½ Kr(2~ Inly)e 2~inx
and
½
inly ) 2~inx
b(n)y Kr,(2~
e
g(z) =
n 0
Suppose that
f(z)
and
g(z)
satisfy
(f(z)IK)(g(z)IK)
where
K
is the operator
L(s,~X~')
= -s~
= f(z)g(z),
g(z) IK - g(-z).
Define for
(s+r+r'
s+r-r'
(s-r+r'] (s-r-r'
~ ) F ( ~ ) F . ~ . F . ~ )
Re(s)
large
~a(n)b(n) n-s-
4F(s)
Then we have
.l+s
n[ 7
,Tx~')
A(s) L ( ~ , w
~ A(s+l)
x w'),
where
A(s) = ~-½sr(~)~(s).
Furthermore
s = I
L(s,~XT')
is regular in
which occurs if and only if
precisely we have
Re(s) > ½
except for a simple pole at
is the contragredient
of
~'.
More
115
co
= ~3( f , g > "
L(s,~X~')
where
(f,g>
_
+ ( f'g)~ +
I c(n)(s-l)n,
n=l
is the Petersson inner product of f and g
and
<f,g>~
is a modified
inner product defined by
<f,g>~ = 6(c -~io$ 2) < f,g) _ 2 ~ / f(z)g(z) log (y61A(z)l)d~,
D(F)
where
A(z)
is the Ramanujan modular form.
Proof.
First observe that because of the parity assumption
(f(z) K)(g(z) K) = f(z)g(z)
we have that
SF = { z = x + i y :
Let
F
= {(
n
i): n C 2Z }
F = SL2(~) /{+12} ;
and let
D(F)
a(-n)b(-n) = a(n)b(n).
Ixl j ½, y ~ 0).
be the fundamental domain for
then we have as in Lemma (2.3)
sr s
On the other hand we have for
Re(s)
Let
~
a e r/r
the formal congruence identity
aD(r).
sufficiently large
/ f (z)g (z)Y sd~
SF
½
= / (/ f(z)g(z)dx)yS-mdy
0 -½
.t 2~i(m-n)xl ~ s-2_
= / ( ~ ~ a(m)b(n)YKr(2~[mly)Kr,(2~In[y)j e
axjy
ay
0
m n
-½
oo
= m~0 a(m)b(m) 0f
Kr(2~]m[y)Kr'(2~Imly)yS-ldy"
The i n t e r c h a n g e of t h e o r d e r of s u m m a t i o n and i n t e g r a t i o n
is justified
by t h e f a c t
116
that uniformly
y ~ ~
in
x
both
(resp. y ÷ 0)
f(z)
and
g(z)
0(y c) (resp. 0(y
are
for some positive constants
We now use the well known identity
c
and
-e'
))
when
e'.
([ 27 ], p. 102)
oo
/ K (C~t)K (~t)t-Pdt
0 P
e~P-I
2P+2F (l-P)
which is valid for
= 2~ny
and
Re(s) > 0
p = l-s
and
Re(I-o±~±V)
> 0.
We apply this identity with
to obtain
ff(z)g(z)ySd~
SF
-s
s+r+r'
~.s+r-r'
s-r+r'
.s-r-r'
r(---f~)~t~)r(T)r~
~)
a(n)b(n)n -s
4F(s)
n=l
= L(s,~ ×~').
On the other hand using the congruence
L(s,~X~')
=
identity for the region
~
/
f(z)g(z)ySd~
D(F)
act/to
/
ac F/F
From the automorphy property of
f(z)
f(z)g(z)ySd~oo
and therefore,
and
f (z)g(z)Y sd~°(7-
D(F)
g(z)
we get
= f(z)g(z)(Im~(z))Sd~
making the change of variable
s+l
s ÷-2'
SF
we have
117
L.l+s
t-~,~x~')
= /
f(z)g(z)E(z,s)da,
D(F)
where
E(z,s) =
~
(ImO(z)) ½(l+s)
O e F/F
is the well known Eisenstein series associated to
tional equation for
E(z,s)
F.
We recall that the func-
is
A(s) E(z,-s)
E(z,s) = A(s+I)
where
A(s) = ~
-½S
representation for
S
F(~)~(s).
This functional equation applied to the integral
L(s,~ x ~')
gives
l+s _
L(--Z-,nx~')
A(s)
= A(s+l) L(
7~
,~x~').
We now recall that the first Kronecker limit formula states ([20 ], p. 273)
E(z,ms-I)
3
~
i +6
s-~
~(c-log
2) - 1
log (y61A(z) i) +
7 e(n)(s-l)n,
n=l
where
oo
A(z) = q-~-(l-qn) 24,
q = exp 2~iz,
n=l
is Ramanujan's modular form.
is regular for
Re(s) > ½
From this expansion and the fact that
except for the pole at
s = 1
E(z,2s-l)
([17 ], p. 44)
the
claim in Theorem (2.7) follows easily.
Remark 2.2.3.
Various other possibilities
convolution method suggest themselves;
for developing Rankin's
among these three are noteworthy of mention.
First the result of Ogg-Winnie Li ([25 ], p. 313)
gives an exact functional
118
equation for an Euler product related in a simple way to
~'
correspond to holomorphic cusp forms of levels
condition that if a prime
in §2.2 below).
is proved for
weight
k
then
L(s,~ × ~')
1
when
and
~
~'
when
~
and
N1
and
N2
qllN1
and
qiiN2 (see example 3
Secondly in Doi and Naganuma [ 8 ]
and level
§2.3 below).
qig.c.d (NI,N2)
L(s,~X~')
satisfying the
an exact functional equation
is associated to a holomorphic cusp form of
is a real analytic cusp form (see example 5 in
The last, and perhaps most attractive of all, is the result of
Jacquet [ 14 ]
where it is shown that
L(s,~ × ~')
or rather a simple multiple of
it, for arbitrary automorphic representations n and ~'
field, satisfies a functional equation.
of
GL2~),
k
a global
It appears that for applications to
arithmetic questions, the result of Jacquet promises to be of much significance,
even in the case of
GL2~AQ).
In a future publication we will persue the problem
of making explicit in the case of
GL2~AQ) , Jacquet's form of Rankin's convolution
method.
2.3.
Some Euler Products and their Functional Equations.
In this section we will give several examples of Euler products that
satisfy relatively simple functional equations.
Example i.
We begin by observing that under the assumptions made in
Theorem (2.7) together with the restrictive condition that
g(z) iK = g(z)
we have, with
% = r
and
f(z) IK = f(z)
%' = r',
L(s,~) = ~ - ½ ( s - % ) F ( ~ ) F - ½ ( s + X ) F ( 2--)s+lii~(l-~pp-S'-l"
I ) ( _~pp---s,-l)
P
= L(I-s,n)
and
and
119
L(s,~') = ~-½(s-%')r(s2%----")~-½(s+%')F(S2%') ~ (l-~'p-S)-l(l-~p-S) -I
P
P
P
=
L(l-s,~').
The multiplicativity of the coefficients
a(n)
and
b(n)
give after an easy
calculation
oo
1
--a(n)b (n)n-s = ~(2s)
-~-det(14-P Sr(gp)®r(gp )) i
n=l
P
1
, -s -i
-- , -s -I
~-i -s -i
- , -s -i
- ~(2s) p~(l-~p~pP ) (l-~p pp ) (l-~p~pp ) (l-~p~pp ) •
Here
r
is the
2-dimensional representation of
is the conjugacy class in
If we now define a
~'
GL2(~)
GL2(~)
containing the matrix
and
{gp} (reap. {g~})_
(0~P0_)~p (reap. (0~
F-factor associated to the automorphic representations
and the complex analytic representation
r@r
~p))0.
~
and
by
F(s,~,~';r@r) =
= ~-½(s-k-l') F (.s-l-k'
T)
-~(s-k+k').~s-k+k'.i<T)~-½(s+k-k')..s+k-k'._-½(s+k+k')r(s+~
k ' - ) ~ t ~ ) ~
then we have
L(s,~X~') =
The functional equation for
A(s) = A(l-s)
L(s,~ ×~')
L(s,~,~';r@r)
4A(2s)
together with the functional equation
now gives the result
L(s,~,~';rSr) = L(l-s,~,~';r~r).
Furthermore, since
L(s,~,~';r@r)
differs from
L(s,~X~')
by the factor
A(2s),
120
it is also holomorphic in the region
if and only if
v
Re(s) > ½
is the contragredient of
and has a simple pole at
~'
or equivalently if
s = 1
z = z[
It
remains an open question to investigate the location of the zeros (and poles) of
the Euler product
L(s,~,~';r®r).
Example 2.
Let
~
and
~'
be automorphic representations of
associated respectively to holomorphic cusp forms of weight
k
of level i.
and
If we put
% =
k-i
--~-'
then the Euler products of
~
and
%,
k-i
%
=-~-'
~'
i-i
o
and
£
GL2~A Q)
and both
%'
2'
~+i
o=--7
-,
have the form
-h(s-%)~.s-L -½(s-%')~s-%'.v-r._ _ -s.-l~. =- -s,-i
L(s,~) = Tf
I~--~-)7T
y~---) 11 [±-%pp ) [--%pp )
P
=
L(I-s,~)
and
L(s,~') =
-½(S-%o)F(~)-½(S-%o)F(~)77-
(l_~;p-S)-i ( l - ~-~
p p ) -s.-i .
P
If we define a F-factor by
4
i=l
where
~+__~k
%1
=
1 -
2
' %2
2
%-k
Z-k
~+k
=
' %3
=
2
' %4
=
2
-
i,
and if we put
L(s,~,~';r@r') = g ( s , ~ , ~ ' ) ~ d e t
P
where
{gp}
(resp.
{g;})
(14-p-sr(gp)®r(gp))-l,
is the conjugacy class in
GL2(~)
containing the
121
0
matrix
(0P ~ o ) ( r e s p .
tation of
GL2(~) ,
o
( P ~.)), and
r
is the standard 2-dimensional represen-
then we have by a simple application of Rankin's convolution
method
L(s,~,~';r@r)
where
= L(l-s,~,~';r@r)"
(-l)mF(s) 2
F(s+m)F(s-m)'
m = k - %.
Remark 2.3.1.
It should be observed that the extra
F-factor appearing
in the functional equation is a simple rational function of s.
Again from the
properties of the Eisenstein series we have that the Euler product
is holomorphic for
s = 1
Re(s) > ½
with the possible exception of a simple pole at
which occurs precisely when
say when
~
is equivalent to
Example
3.
L(s,~,~';r@r)
~
is the contragredient of
7',
that is to
~'.
This example, due to
Ogg and Winnie Li ([ 25 ], P. 313),
deals with two automorphic representations which may be ramified but whose conductors satisfy certain arithmetical properties.
representation of
conductor
N1
GL2~A~)
and
trivial central character; let
GL2~A Q)
k,
and trivial character.
N2
product of all primes
that for every prime
q
q
~
be an automorphic
associated to a holomorphic cusp form of weight
representation of
conductor
Let
7'
be another automorphic
associated to a holomorphic cusp form
that divide
which divides
M
M
Let
g(z)
M = ~.c.m. (NI,N 2)
and
and for which ordqN 1 = ordqN 2.
N,
f[[Vq ]k = ~qf'
ordqN 1 = ordqN 2 = 1
gl[v_M]kq = nqg
with
~2 = 2
q
Nq = I,
k,
and
of weight
N
is the
Suppose
122
where
Vq
and
x~y,z
M
q x yq ) ' ~ = ordq M
= (Mz
q
are integers satisfying
2~
x - yMz = q .
If we define a
P-factor
by
P(s,~,~';r®r)
4
= ~-2(s-%i)F(~),
i=l
where
%1 = l-k, %2 = -k, %3 = 0, X4 = -1
and if we put
L(s,~,~';r®r)
= F(s,~,~';r®r) (1-~qnqq-S)-l~det(l.-p-Sr(g~ql
II'N
) 8r(g'))
-Ip
P~M
4
p
v -i ~"(det(12-q-Sr(gp))-l~"'det(12-q-Sr(gq'))
• ~ ' ( l - q -s ~q~q)
-I,
where
L(S,~) = P(s,~) ~
(l-Eqq-S)-i V
-s -i
-- -s -i
qlNl
P~Nl(l-Epp ) (l-Epp )
and
L(s ) =
(lqqS>1 V (l ppS) l(l%pS) -1
q IN2
are the Euler products of
the conjugacy classes in
~
and
GL2(~)
P~N 2
~'
respectively and
{gp}, { gp'} ,{ gp'} ,{gp,,}
that contain respectively the matrices
are
123
0
~'
(~p ~p) , (0p _~,) , [~q~ 0q~q) ' <0
f~q~ ~'-~0~q~q]
and the products
(NI,N2),
N2
~',
but not
~",
N1
~"',
and
run respectively over the primes that divide
N1
L(s,7,7';r®r)
Furthermore,
L(s,~,~';r @ r)
simple pole at
s = 1
if
but not
= M
N 2.
~--SL(l-s,7,~ ;r®r).
is holomorphic for
7
We then have
Re(s) > ½
is the contragredient of
the equivalence of the local components of
7 = ~7
with a possible
7'.
and
The requirement about
7' = 57'
P
P
at the infinite
prime can be relaxed in the proof of the above functional equation by making some
trivial changes in the argument in [ 25 ].
Example 4.
The following example, whose significance is best understood
in the contest of the theory of automorphic representations for
obtained by Shimura ([ 38 ]).
GL3~A),
was
The method of proof generalizes in a non-trivial
way Rankin's convolution method to automorphic forms on the Metaplectic group.
Let
7 = @~Tp be an automorphic representation of
morphic cusp form of weight
k
and level
I.
7 .
P
2-dimensional representation of
r2
We define a
and
Let
= ~ 7-½(s-li) F ( ~ )
i=l
with
l I = 0, ~2 = -i,
and if we put
rI
p
let
{gp}
13 = k - 2
be
be the standard
the symmetric square of
F-factor by
F(s'7'r2)
associated to a holo-
For each prime
the semisimple conjugaey class associated to
GL2(¢)
GL2~A ~)
r I.
124
L(s,~,r2) = r(s,~,r2)~det(13-p-Sr2(gp)) -I,
P
then Shimura's result, which is valid for a more general automorphic representation
7,
is that
L(s,~,r2) = L(l-s,7,~2)
and, most important of all,
proved that
tation of
L(s,~,r)
GL3~A~) ,
Example 5.
L(s,~,rT)
is entire.
Gelbart and Jacquet [ ii ]
is in fact the Euler product of an automorphic represenin accordance with
Langlands' functoriality principle.
This example, due to Doi and Naganuma [ 8 ],
of the earliest evidence of how an automorphic representation
could be lifted to an automorphic representation of
quadratic field.
have
Let
7 = ~p
GL2~A F)
~
provided some
of
with
GL2(/A~)
F
be an automorphic representation of
a real
GL2~A Q)
whose component at the infinite prime belongs to the holomorphic discrete series
and is unramified at all local components, i.e.
cusp form of weight
of
GL2~A~)
k
and level
i.
Let
7'
7
is associated to a holomorphic
be an automorphic representation
associated to a real analytic wave form in
Let the Euler products of
~
and
7'
Wg(N,%),
with
~ =
l-r
2 "
be respectively
L(s,~) = r(s,z,r)~det(12-p-Sr(gp)) -I
P
and
L(s,~') = r(s,z',r)
where
and
by
~q
r
q~N (l-6qq-S)-ip~N det(12-P
is a complex number and {gp}, {g~}
r(gp))-i
are conjugacy classes in
is the standard 2-dimensional representation of
GL2(~).
GL2(K)
Define a r-factor
125
F(s,~,~';r@r)
= ~~-]H~-½(s-ki)F(~ii)
i=l
with
k-I
Ii =
k+l
2
r, 12
2
k-I
r, ~3 =
2
k+l
+ r, ~4 = - - - +2
r~
and put
L(s,~,~';r®r)
=
-i
- q-Sr(~qgq))-ip~Ndet(l 4 - p-Sr(gp) ® r(gp) )
= F ( s , z , ~ ' ; r @ r ) ~2d eNqt (ll
We then have that
L(s,~';r®r)
L(s,~,~';r@r)
Here
W(~,~')
= W(~,~')N2(½-S)L(l-s,{,@';r@r).
is a complex number of absolute value 1
subsequent discussion.
7'
is holomorphic and satisfies
which plays no role in our
To obtain the result of Doi and Naganuma, one takes for
an automorphic representation associated to a 2 dimensional semisimple repre-
sentation of the Weil group of a real quadratic extension
F
of the rationals
which is induced by an unramified grossencharacter of the idele class group
of
F.
Example 6.
The following example was first pointed out by Langlands and
was suggested by his theory of Eisenstein series.
Due to our lack of information
about ramification phenomena, we consider only the simplest situation corresponding
to an automorphic representation
~ = ~
of
GL2~A ~)
of conductor 1
associated
P
to a holomorphic cusp form of weight k.
standard representation of
in
GL2(~)
GL2(~).
Let
Let
r3
{gp}
attached to the local component
be the symmetric cube of the
be the semisimple conjugacy class
~ .
p
Define a F-factor by
126
= ~
F(s,~,r3)
~-;~(s'li) F (q--),
s-li
i=1
where
4
det(l 4 - r3(X )T) = ~ (i - %iT),
i=l
and
{Xoo}
is the semisimple
conjugacy class corresponding
~ .
Define the Euler product of the Data {~,r 3}
as in
to the local component
§1.4
-s
L(s,~,r3)
= F(s,~,r3)~det(l
P
4 - p
by
-i
r3(gp))
•
We then have
L(s,~,r3)
where the constant g(~,r3)
is independent
The proof of this functional
the examination
= 8(~,r3)L(l-s,~,r3) ,
Example 7.
s
and plays no role in our results.
equation is given in Shahidi
of a 'non-constant
exceptional Lie group
of
term'
Euler products obtained b ~ base change.
consider only an automorphic
to a holomorphic
G
~
of conductor
ard 2-dimensional
the Frobenius
class in
and let
f(o)
and let
representation
associated
jugacy class of a generator
K ~ .
given by
~ = 97
cusp form of weight
G = GaI(K/~).
Let
X = trace 0-
P
k.
[ 22 ]
of
Let
0: G + GL(V)
Let
For simplicity we
GL2~A~)~
K
be a representation
GL2(~).
For a finite prime
G
and let
{gp}
Gal(K ~R)
w
where
Recall that the r-factor associated
of conductor i
be a finite Galois
of
p
be the semisimple
K
~ .
P
w
con-
of the Saito-
r: GL2(~) ÷ GL(W)
to the local component
of
The following
representations.
representation
eonjugacy class in
GL2(~)
series of the
G 2.
Shintani theory of liftings of automorphic
extension of
and depends on
of a certain Eisenstein
jectural example is suggested by Langlands'reformulation
associated
[ 36 ]
Let
{~ }
of
be the standlet
~
P
be
conjugacy
be the con-
is a simple factor of
to the Artin L-function L(s,O)
is
127
dimo
~.
~
,,
•
.
r(s,~) =7T ~-~ts-~r(~)
i=l
with
hi =
Recall that the
0
if
I < i < X(1)+X(O°°)
-i
if
)<(1)+X(°°°)
< i -< dimo.
2
F-factor of the Euler product
F(s,~,r)
2
=~j=l
L(s,~,r)
is
~
~o,
~o
2(s-aj) r(s-!J)
2
with
o =
k-i
o
hl
- --2-' %2
We define a new F-factor associated
r(s,~;r®p)
k+l
2 "
to the data
{~,r,p)
by
2 dimp h .
.o ~ ,
4o
= ~T ~
7-21S-Aj--Ai) r(S-A$--Ai).
j=l i=l
2
If we put
L(s,~;r®p)
= £(s,~;r@p)~det(I
- p-Sr(gp)@plVIp(Op))-l,
P
where
VIp
is the subspace of invariants
of
V
of the inertia group
Ip, then
it is to be expected that
L(s,z;r@p)
where
dent of
~
= £(~,r®p)f(0)2(½-S)L(l-s,~;r®P),
is the contragredient
s
of
and of absolute value i.
representation
of
G
and
p
and
g(~,rSp)
is a 'root number'
If this is true and if
PK
indepen-
is the regular
128
PK =
is its natural decomposition
[~ (dim@)~
~eG
then the Euler product
L(s,~;r®PK) = ~^L(s,~;r®@) dim@
~eG
would satisfy the functional
equation
L(s,~;r@PK ) = £K(~,r)d~(!~-S)L(l-s,~;~®~K),
where
dK
is the absolute value of the discriminant
functoriality
automorphic
principle
suggests
representation
of
that
GL2~)
Shintany theory as given in [22 ]
If one is only interested,
L(s,n;r®p K)
;
K/~.
as we are in these notes,
is 2-dimensional
of the Euler product
L(s,~;r~PK),
decisive
representation
get the analytic continuation
representations
necessarily
then the Saito-
information
~
L(s,~;r®p)
n(p)
of
of the Euler product
9-
The simplest
Using Rankin's
and in some cases the
L(s,p)
GL2~A~).
such situation
that
is the Euler
This allows us to
L(s,~;r@O K)
which may be ramified and number fields
abelian over
about the
under the assumption
and its associated Artin L-function
product of an automorphic
Q.
in the analytic continuation
convolution method we can obtain the analytic continuation,
p
k is cyclic over
nature of the Euler product for abelian extensions.
equation,
Langlands'
is the Euler product of an
in the special case when
Shintani theory also proves this and provides
functional
In fact
this is indeed proved by the Saito-
and the functional equation of the Euler product
holomorphic
of
K
for automorphic
which are not
arises when
K
is the
splitting field of a cubic polynomial with integer coefficients
whose discriminant
is not a square,
group
in which case
whose representations
Example 8.
GaI(K/~)
is the full symmetric
S 3,
all of
are monomial.
An Euler product that contains
two variables.
Let z
be an
129
unramified automorphic representation of
analytic cusp form
f(z).
GL2~A ~)
which is associated to a real
Let
f(z) =
[ a(n)y2K%(2~Inly)exp(2~inx)
n#O
be its Fourier expansion about the cusp at infinity with
a(1) = i.
Let
E(s,z)
be the Eisenstein series of
§2.1.
functions
o (n)
s
and Rankin's convolution method show that for complex
variables
s
and
and
a(n)
s'
The multiplicative property of the arithmetical
the expression
L ( s ' , s ; ~ E)
=
/
E(z,s)f(z)S(z,s')d~
Do(F)
-½(s'+l)
s ~2s+l
r
s 4J~s-I
s --~s+~
(----if----) r (---y--) r ( T )
s - 2s+~
s --~s-I
4r (s') A (s+l)
is an Euler product.
n~oOs~Inl)
I t? a-~)
The functional equation for the Eisenstein series
gives readily the following functional equation for
..l+s'
c~--y-,s;~ E)
=
A(s')A(s)
A(s'+l)A(s+l)
In fact a direct computation shows that if
automorphic representation
L ( l+s'+s
~ , ~ ) L ( - - - --l+s'-s
~--,~)
Example 9.
~-(s,+½s)
r (---y--) r (-----if----)
7,
then
E(z,s)
L(s',s;~®~ E)
l-s'
L(--~-,-s;~e~E)
L(s,~)
is the Euler product of the
..l+s'
L[--~--,S;~T E)
is actually equal to
times a simple expression that involves only A(s).
Let
S
be the symplectic group of rank 2
and let
~ = ~
P4
P
be an automorphic representation of
components.
For a finite prime
in the dual group
S05(E).
p
Sp4~A ~)
let
{gp}
which is unramified at all the local
be the semisimple conjugacy class
Assume that the component
~
belongs to the
130
holomorphic
discrete series of 'weight'
modular form of weight
Hecke operators.
S05(~).
Let
k
k,
that is
~
is generated by a Siegel
which is a cusp form and an eigenfunction
r
be the natural
4-dimensional
for all the
complex representation
of
Define a F-factor by
4
= ~-½(s-%i)r(s-%i),
i=l
2
r(s,~,r)
with
~I = k-i
2'
~2 - k-i
2
i, %3 = -%1' ~4 = -~2"
If we put
L(s,n,r)
then Andrianov
[ 1 ]
= F(s,~,r)~det(l
P
has shown that
L(s,n,r)
It is also shown that
be observed
that
4 - p-Sr(g ) ) - i
P
L(s,~,r)
L(½,~,r)
=0
= (-l)kL(l-s,~,r).
has at most a finite number of poles.
for
occurs for classical automorphic
k
an odd integer,
forms on the group
It should
a phenomenon which already
Sp2
=
SL 2
of weight
k s 2
mod 4.
2.4.
The Average Size of the Eigenvalues
of Hecke Operators.
In this section we obtain information
that appear in the Fourier expansions
be used to determine
L(s,~);
of automorphic
forms on
GL2~A ~)
which can
the location of the critical strip for the Euler products
we also get some information
correspond
about the size of the coefficients
to the unramified
about the zonal spherical
local components
~
P
functions
of an automorphic
that
representation
131
= ~
. We first prove a simple result that suggests that the Petersson-Ramanujan
P
conjecture is true on the average for real analytic cusp forms.
Theorem 2.8.
Let
~
be an automorphic representation of
is associated to a real analytic wave form
f(z)
in
GL2~A ~) which
Wx(N,%), % = ½(l-r2).
Let
its corresponding zeta be
oo
~(S,Z) =
[ a(n)n -s
n=l
= q~N(l-a (q) q-S)-ip~N(l-a (p) p-S+x(p) p-2S)-i.
Then we have for
x
a positive real number
la(n) I2 = c(~)x + o(x),
n<x
where
3
<f,f>
1
c(Z) = ~'F(½+r)F(½ r ) ' N ~ ( l + p
i)"
pIN
Proof.
of
Fo(N)
Let the Fourier expansion of
f(z)
about the cusp at infinity
be
f(z) =
~ a(m)y=Kr(2~Imly)exp(2~imx).
m#0
The modified Bessel function
I
Ks(Z) = 2 "
where
-S
(z) - Is(Z) '
sin(s~)
132
(½z) s+2m
I(z) = m ~ 0 ~ l )
is clearly real valued when
valued and
s
and
s is pure imaginary.
z
are both real and also when
z
is real
First we note that
~½
/ / If (z) I2ySd~
o -½
oo
½
-~
½
= f yS
a(m)y Kr(2~Imly) ~ a(n)yaKr(2~Inly) / e2~ix(m-n)dx'y-2dy
m#0
n#0
-½
o
co
eo
= 2 ~ la(m)I 2 /oKr(2~my)Kr(2~my)yS-ldy.
m=l
Again we use the identity
oo
/ K~(~t)Kv(~t) t-Odt
o
aP-i
l-p+B+v
2P+2F(I-p)
with
~ = 2~n,
p = 1 - s
l-p+p-v
l-p-p+v
and
p = ~ = r
to get
~½
/ f If( z ) 12ySd~
o -½
(4~)-sF(~)F(~)F(2)2
4F(s)
co~ la(n) 12n-s
n=l
= L(S,~ x~).
Recall that if
l-p-~-v
.r(----y--)C(----F--)r(~)r(----f----)
Foo = {(i0 i): n ~ 7z}, S
S = {z = x + i y :
is the region
Ixl __<½, y > 0}
133
and
D (N)
o
is a fundamental domain for
r (N)
o
then
S E Z ~Do(N),
where
modulo
the sum
r .
E
runs over a complete set of coset representatives
We can therefore
integral expression for
replace the region
L(s,~X~)
L(s,~X~)
S
If(z) 12Y sd~
o ao (N)
o
= ~ /
If(z) 12ySd~°o.
0 D (N)
o
The automorphy property of
f(z)
f.az+b.
[c--z-~) = x(d)f(z)'
and the
SL2-invariance
of the measure
(a b
e d ) C Fo(N )
d~ = y-2dxdy
imply
If(z) I2ySd~o~ = If(z) 12(Im~(z))Sd~.
This then gives that
L(s,~X~)
= /
If(z)[2E(z,s)d~,
D (N)
o
where
S(z,s)
= [amo(~))
O
= ySFN(Z,S )
s
r (N)
o
by its equivalent in the
to obtain
= ~ I
of
is the Eisenstein series for the group
infinity.
Fo(N)
corresponding to the cusp at
We now investigate the constant term of
observe that if
F
E(z,s)
more closely.
First
is the stability group of the cusp at infinity then an
equality of cosets
a b
a'
b'
F~( c d ) = F (c, d , )
occurs if and only if
(c,d) = ±(c' d'),
so the left cosets of
in i-I correspondence with the pairs (c,d);
atives the pairs
(0,i)
and
(c,d)
with
hence we may choose
c > O,
Nlc
and
then
ImO(z) = icz+dl2,
Y
z = x+iy.
Thus
FN(Z,S ) = yS +
ySlmNz+nl-2s"
~
m>0
(mN,n)=l
Let
-I
~N(S) =
n>0~ n-S = p~N (l-p-s)
(n,N)=l
Then we have
2~N(S) FN(Z, s) =
~
ImNz+n 1-2s
m,n
(n,N)=l
)~'t,~=+~l -ms Z ~(d)
m,n
dln,N
d>0
in
as
represent-
(c,d) = i.
that if
O(z) = az+b
cz+d
F
F (N) are
o
Recall
135
=
[ B(d>
diN
[
]mNz+n1-2s
m,n
= d~N~(d) d-2SG(~, s) ,
where
(m,n)
~
is the Mobius function, the sum E'
m,n
different from (0,0) and
G(z,s)
is an Epstein zeta function.
=
runs over all pairs of integers
~' Imz+n1-2s
m,n
We use the fact
E(z,s) = ySG(z's)
2~(2s)
=
Z
(Im°(z)) s,
o 6 F/F
where
series.
F
is the unimodular group
SL2(~ ) /{±12 }
and
E(z,s)
Thus we get
YS p ~N (1-p-2S)FN (z,s) = N-Sd~ND(d)d-SE(~,s).
Let
E(z,S,Fo(N)) =
[
(ImO(z)) ½(s+l)
e F (N)Ir
o
and
E(z,s,F) =
[
o6F
(ImO(z)) ½(s+l)
Foo
is its Eisenstein
136
denote respectively
r (N)
o
and
the Eisenstein series for the cusp at infinity for the groups
F-(observe the change of variable
s ÷ ½(s+l)!).
We then have the
identity
E(z's'F°(N))
= pIN~(l-p-l-s)-i d~N p(d)(Nd)-h2(I+s)E(~'s;F)"
The well known Fourier expansion of
easy change of variable
E(z,s,F) = y½(l+s)+
where
Os(n)
A(s) = ~
-½s
([17 ], p. 46)
in the simple
can be put, after an
form
l~1½s .y½K~
=s (2~Imly)exp(2~imx)'
is the sum of the
and
E(z,s,F)
A(s) ½(l-s), ~
2
Os (Iml)
A(~
y
tm$ 0 ^(s+l)
s
F(~)~(s)
E(z,S,Fo(N))
s+l
s +--~--,
s-th
powers of the positive divisors of
is Riemann's Euler product.
E(z,s,F)
we substitute
n
and
In the identity relating
this last Fourier series to obtain,
after rearranging the terms involving the Mobius function
p(d),
E(z,s,F o(N)) = y½(l+S)+c(s)y2(l-S)+m~0Cm(S)y2K½s(27[m[y)exp(2~imx),
where
A(s) ~I(N)
c(s) = A(s+l) }s+l(N)'
~a(N) = N a T ( 1 - p
PIN
and the coefficients
c (s)
m
are all holomorphic
To obtain the residue of
several ways.
-a )
E(z,S,Fo(N))
in the region
at
s=l
The easiest is to evaluate the residue of
interesting way is to appeal to Kronecker's
Re(s) > 0o
we can proceed in
c(s).
A somewhat more
limit formula ([20 ],p° 273)
for the
137
Eisenstein series
E(z,s)
which in a neighborhood of
s = 1
can be written as
1
12
k
2)
6
E(z,s,F) . . . . .s-i + - 7 (Y - log 2 - log y2]q(z)]
+
%T
and then substitute into the identity which relates
E(z,s,F)
0(Is-il)
to
E(z,S,Fo(N)).
We then obtain
E(z,s,Fo(N)) = U (l-p-l-s)-i ~ p(d) (Nd) -½(s+l)
pIN
diN
6
x (~- s_--ll+ (y - log 2 - log (N--Xd)½[~(~)I2) + 0(Is-ll)),
and this is
= 6.
1
.i
~T N g ( l + p I-) s-i
We can even compute the constant
A(z)
+ A(z) + 0(Is-ll)
in the Laurent expansion by using the
appropriate terms in Kronecker's limit formula.
to obtain, again after a change of variable
We put together the above results
s+l
~
+ s,
oo
L(s,~X~)
3
< f,f>
= ~'Ng(l~)
A comparison of the poles of
.i
+ !0an.(S_l)n"
s-i
n
E(z,s,Fo(N))
and using the identity
(4~) -Sr (s~--r)r (s~--r)F (2) 2
la(n) 12n -s = L(s,~X~)
4F(s)
shows that
r
n= 1
cannot be real, and in particular
the F-factors one obtains
r # ½.
Therefore dividing by
138
n=l
la(n) 12n-S = 3__ •
( f,f )
73
F(½+r)F (½-r)
1
Np~N(l+p-l)
_i_l + [ bn(S_l)n"
s-i
n=0
A standard application of the Wiener-Ikehara Theorem to the above Dirichlet
series gives
la(n) l2 = ~ .
n<x
( f,f >
F (½+r) F (½-r)
• x
+
o (x).
NyN(I+p-I)
This completes the proof of Theorem 2.8.
The Cauchy-Schwarz inequality and the above asymptotic estimate give the
following corollary.
Corollary 2.9.
With the notation and assumptions as in Theroem 2.8.
have
la(n) l << x.
n<x
To obtain estimates which are more precise than those of Theorem (2.8)
one must use the full strength of the functional equation for
this we proceed to do now.
Theorem 2.10.
(I)
Suppose the following six conditions are satisfied:
BI,...,~4, 61 .... ,64
are positive real numbers and
~i,...,~4, yl,...,y4
= Y1 + "'" + Y4 - ~i - "'" - ~4 > ½;
c(n) > 0
and
and
First we recall Landau's Theorem [ 18 ]:
are real numbers satisfying the inequality
(II)
L(s,~ x ~),
we
139
co
Z(s) =
[ c(n)n -s
n=l
is absolutely
(III)
convergent
The function
for
Z(s)
Re(s) > B
and represents
has a meromorphic
plane and in each fixed strip
o I < o < 02
there a regular function;
continuation
to the whole complex
it has at most a finite number of
poles;
(IV)
for some
A > 0
co
F(~I+BIS)...F(~4+B4s)Z(s)
= F(yi-61s)...F(y4-64s ) [ e(n)(An) s,
n=l
the last sum being absolutely convergent
(V)
Re(s) < 0;
Z(s) = 0(e Yltl)
for large
(VI)
for
Itl
and some constant
for some constant
y = Y ( O l , o 2)
in any strip
o I < O < 02;
B > 0
le(n) InB = 0(xB(log x)B).
n<x
Then we have
c(x) =
~ c(n) = a(x) + 0(xK(log x)g),
n<x
where
g = max (B,m(B)-I),
m(B)
is the multiplicity
of the pole of
Z(s)
at s = B,
2D-I
K = B" 2N+I'
and
R(x),
(x > 0),
in the strip
is the sum of the residues of
xSz(s)
- - a t
the poles of
~ < ~ < ~.
To apply Landau's
theorem we show first that the series
Z(s)
140
Z(s) = ~(2s)
~
la(n) 12n -s
n=l
co
= [ c(n)n -s
n=l
satisfies
the c o n d i t i o n s
a2 = -r,
63 = a4 = 0,
of the theorem.
We take
~i = 6i = ½'
T 1 = ½ + r, T2 = ½ - r, T3 = T 4 = ½
= Y1 +
i < i < 4, 61 = r,
and o b s e r v e
that
"'" + Y4 - ~ i - "'" - ~4
=2>½.
By T h e o r e m (2.8)
we have
C(x) =
~ c(n)
n<x
la(~) I 2
p>
2
< x
~
la(P) I2
0(~)
= 0(x).
½
=
~
I)< x ½
Therefore
by partial
summation
we observe
Z(s)
=
that
oo
~ c ( n ) n -s
n=l
is a b s o l u t e l y
Eisenstein
function
convergent
series we know
for
that
Re(s)
> 1
and h e n c e
( [ 1 7 ], p. 43)
~ = i.
at a p o l e of
F r o m the t h e o r y of
&(2s-l)
A(2s)
the
'
t41
A(2s)E(z,s)
A(2s-l)
is holomorphic.
Furthermore
each fixed strip
A(2s)E(z,s)
o I < O < ~2"
Hence condition
The well known behavior of the
fact that for
Itl
sufficiently
A(2s)E(s,z)
has at most a finite number of poles in
(III) is satisfied.
F-function on vertical
strips and the
large
= A(2s)y s + A(2s-l)y l-s + 0(e -cy)
imply the estimate
Z(s) = 0(eYltl),
as required by condition
Condition
by theorem
(VI)
(V).
is simply satisfied because
e(n) = c(n)/~2n-
and hence
(2. 8)
e(n)n = 0(x).
n<x
Now
where
3
K = ~
and
the only pole of
Z(s)
in the strip
~5 -< ~ j 1
the residue of
Residue
s=l
Landau's
xSR(s)
s
3
~2
(f,f >
F(½+r)F(½-r)
theorem then gives the estimate
3
c(n) = Co(~)x + O(x5).
n<x
X.
is at
s = 1
142
Let us now recall that
la(n)
12n-s
=
n=l
~ c(n)n -s ~ p(m)m -2s
n=l
m=l
and therefore
la(n) l2 =
~
c(k)p(h);
kh2=n
we then have
la(n>l 2 =
n<_x
c(k)p(h)
Z
kh2<x
=
~
h<x½
-
-
>(h)
~ c(k)
k<-x
--h2
h<x½
h
c (7) x
h<x ~
3
h
but
Co (~)Xh!x~~h)h2 : Co<~)x{~ + 0(x%}
and the error term gives
3
o( x5
6
3
Z ½h5) = 0(x5) •
h<x
The final result is
Theorem 2.11.
have that
With the notation and assumptions as in theorem 2.8., we
143
3
[ la(n)1 2 = c(T[)x + 0(x~).
n<x
If in the above theorem we replace
x
by
N
and
N-I
and subtract
the
two expressions we find that
3
la(N) 12 << N~
or
3
la(N) l << N I0,
where the implied constant depends on the automorphic
state this result in group theoretic
Theorem 2.12.
with
z
Let
w = @~T
P
a member of the principal
for which
~
be an automorphic
series.
Let us
representation
of
GL2~A ¢)
Then for all those finite primes
is a class one representation
~
z.
terms.
either
P
series or
representation
~
belongs
p
to the principal
P
belongs
to the complementary
series
~(~i,~2)
with grossenchar-
P
acters
p1
and
P2
g i v e n by
h(x) = Ixl%
and
h(x) = ,~,-°
3
0 < o < TO"
3
Proof.
la(N)[ << N I°
From the estimate
1
=
1 - a(p)T + X(p)T 2
has radius of convergence
r
with
r J~0"
it follows that the power series
~ a(pn)Tn
n=0
Therefore by Gelbart
([ i0 ] p. 72)
144
3
la(p) l = Ip°P + p-°P I <__ 2pI--~
and hence the estimate
0<(7
--
Remark 2.4.1.
that if
~ = ~p
3
<-i0"
p
--
It has been conjectured
is an automorphic
a finite number primes
p
([12 ], P. 357 and [33 ], P- 264)
representation
the local components
of
z
GL2~A~)
then for all but
belong to the principal
P
series.
= ST
A more comprehensive
is an automorphic
conjecture
of Langlands
representation
of
~A'
([24 ], p. 56)
is that if
then for all primes
p
the
P
character of the representation
~
is a tempered distribution.
If
~ = ~
P
is an automorphic
representation
morphic discrete series,
ture establishes
of
P
GL2~A~)
with
z
a member of the holo-
then Deligne's proof of the Petersson-Ramanujan
this; if
z
is not amember
of the holomorphic
then the above results suggest that this conjecture
discrete
conjecseries,
is indeed true, at least on
the average.
The earlier results of Rankin
of the coefficients
given by Theorem
of a holomorphic
(2.8)
Theorem 2.12.
and let
L(s,Z)
Let
about the average size
cusp form of arbitrary
level and the estimate
imply the following result.
Let
z
be an automorphic
be its associated Euler product.
located in the strip
Proof.
([ 30 ], p. 357)
0 < Re(s) < i,
z = ~
i.e.
representation
GL2~A~)
Then the zeros of
the critical strip is
and disregard
of
L(s,Z)
are
0 < Re(s) < i.
the finite number of local represen-
P
tations which are not class one and construct
the remaining
local factors.
the modified
zeta
~(s,Z)
with
Then
co
1
~(s,Z)
~'(i
p
- a(p)p -s + X(p)p -2s) =
[ A(n)gz(n)n
n=l
s,
where the prime in the product denotes that only those local factors enter which
arise from class one representations.
of the coefficients
a(n)
and
A(n)
By Theorem ( 2 . 8 )
and the multiplicativity
we obtain easily the estimate,
for large
x,
145
IA(n)~(n)l
<<
x.
n<x
Hence the series and also the product converge absolutely
this implies that
~(s,~) -I
connot happen unless
represents
~(s,~)
the Euler product itself
one uses the properties
with
known estimates for
q
of the
does not vanish in
dividing
([25 ], p. 295)
= ~
is an automorphic
equation.
§i.
The
for these one uses the
representations
series.
L(s,~)
of
GL2~A ~)
have no zeros on the lines
with the following
representation
of
is
Ina latter section we
This will in turn lead to asymptotic
We end
Re(s) < 0
all have the form
to those that are common in the study of the distribution
Remark 2.4.3.
and
that the location of the critical
of Eisenstein
shall prove that in fact the Euler products
Re(s) = O.
~;
To see that
This proves the theorem.
It is indeed remarkable
governed by the general properties
and
~(s,n)
this
to show that these local factors do
strip for the Euler products of automorphic
Re(s) = 1
in
the conductor of
not vanish outside the critical strip.
Remark 2.4.2.
Re(s) > 1
F-function and the functional
and
Re(s) > if
does not vanish in the same region.
L(s,~)
a(q)
Re(s) > 1
a regular function in
finite number of local factors not considered
(i - a(q)q -s)
for
GL2~A ~)
estimates
similar
of prime numbers.
important observation.
and if its associted
If
zeta
P
~(s,~),
that is the Euler product of
~
without
the
F-factors,
has a Dirichlet
series expansion of the form
~(s,~) =
~ a(n)n -s
n=l
and if all but a finite number of the local components
series then for large
x
la(n) l << x,
n<x
belong to the principal
P
146
w h e r e the implied constant is independent of
stant w i l l w o r k for
all
7.
(2.8)
in the sense that the same con-
This is indeed the case for a
a m e m b e r of the h o l o m o r p h i c discrete series.
from Theorem
~
~ = 8z
p
with
In general the estimates o b t a i n e d
only give the b o u n d
]a(n) l << ll~llx,
n<x
w h e r e the inplied constant is independent of
w h i e h m e a s u r e s the global size
metric
of
~
but w h e r e
II~II
is a constant
% and is a constant m u l t i p l e of the P e t e r s s o n
< f,f > of a n o r m a l i z e d a u t o m o r p h i c form a s s o c i a t e d to
~.
All of our
subsequent estimates w i l l be effective only in so far as we are w i l l i n g to carry
the constant
representation
II~II
~.
as an independent parameter among those that c h a r a c t e r i z e the
147
§3.
3.1.
Zeros in the Critical Strip.
The Hadamard Product Formula.
We begin now the study of the distribution of zeros in the critical strip
of the Euler products
GL2~A~).
L(s,~)
associated to automorphic representations
The methods, worked out here only for GL2,
~
of
are quite general and apply
to the Euler products of other adele groups over algebraic number fields.
In this
more general setting we have investigated some analogues of the Brauer-Siegel
Theorem
in the theory of Artin's L-functions.
Throughout this section we fix an automorphic representation
GL2~A ~)
and denote, as usual, by
L(s,~)
its Euler product.
Hadamard factorization theory to the entire function
that there exist positive constants
c
and
c'
L(s,~)
such that
~
of
To apply the
we must first verify
L(s,~)
satisfies the
growth condition
(1)
for
(2)
L(s,~) = O(exp cls I log Isl)
Isl
sufficiently large, but the weaker condition
L(s,~) = 0(exp c'Is I)
does not hold for all sufficiently large
Isl.
We first decompose the Euler product
L(s,Z)
into its
F-factor and the
product of all local factors at the finite primes:
L(s,~)
To investigate the growth behavior of
= r(s,~
~(s,~)
)~(s,~).
we consider its associated Lindel~f
148
function
p(O,z).
of numbers
~
Recall that for each
0
p(O,z)
is defined as the lower b o u n d
such that
~(~ + it,~)
o(ItI$).
It is w e l l k n o w n f r o m the general theory of Dirichlet series that, as a f u n c t i o n
of
~,
~(~,~)
is continuous,
that no arc of the curve
non-increasing,
y = ~(~,~)
and convex downward in the sense
has any point above its cord; also
B(~,~)
is never negative.
We have a l r e a d y indicated in
§2.4
that
¢(s,~)
gent, as an Euler product or as a Dirichlet series,
~(s,~)
is b o u n d e d for
o > 1 + 6(6 > 0),
p(o,~)
for
is a b s o l u t e l y converRe(s)
> i;
in p a r t i c u l a r
and therefore
= O, ~ > i;
from the estimate
r(l-s,~)
F(s,~)
where
= max(I~lI,I~21),
0((It I +
I~l) 2(~-½))
and the functional e q u a t i o n of
L(s,~)
it follows
that
p(o,~)
= 2(½
- 5),
by c o n t i n u i t y these equations also hold for
The cord j o i n i n g the points
is
y = i - ~;
(0,i)
o < 0;
o = i
and
and
(i,0)
~ = 0
on the curve
from the convexity p r o p e r t y it then follows that
p(o,~) < i - ~,
0 < o < i,
respectively.
y = ~(~,~)
149
and in particular
~(½,~) J ½, that is
~(½ + it,~) = O([t[ ½+£)
for every positive
between
D(O,~)
0
and
i.
consists of
~.
The exact value of
is not known for any
2
straight lines:
More generally,
if
for
~
~ < ½
and
~(o,~) = 0
Hypothesis for the factor
for
o > ½.
is an automorphic representation of
is a finite dimensional complex representation of
§1.4
o
An analogue of the Lindelof Hypothesis is that the graph of
~(o,~) = (½ - 4)2
r
~(~,~)
~(s,~,r)
GL2(~)
in the Euler product
GL2~A ~)
and
then the Lindelof
L(s,~,r)
defined in
is that the graph of the associated Lindelof function consists of
2 straight
lines:
~(o,~,r) = (½ - o)dim r
for
O < ½
and
~(O,~,r) = 0
for
~ > ½.
By Stirling's formula we know that
F(s,~) = 0(exp c[s] log Is]), Re(s) ~ ½,
with
F(s,~)
c
a positive constant.
and
~(s,~)
The function equation and the above estimates for
give
IL<s,~)l = O<e=p clsl ~og I s l ) .
This proves
(i).
If we now restrict
s
to the real axis and put
s = ~ > ~
--
then
O
150
Stirting's
formula gives
I£(s,~)l ~ exp aO log o
for some positive constant
a.
Fourier-Bessel
of the automorphic
coefficients
On the other hand the polynomial
form associated
growth of the
to
~
imply that
oo
l~(O,v) l = I ~ a(n)n-Sl
n=l
co
> 1 -
Z la(n)In -~
n=2
>½
for
o
sufficiently
say
o _> Go,
large.
Therefore,
for large real positive values of
~,
we have
IL(o,~) I > exp c'O log o,
where
c'
is a positive constant.
We have thus verified
Hadamard Factorization
conclusions
that
L(s,~)
(2)
cannot hold for all
satisfies all the requirements
Isl.
of the
for subsequent use we now collect the resulting
in the following theorem.
Theorem 3.1.
(non-trivial)
ii)
Theorem;
This proves that
i)
The Euler product
zeros in the critical strip
As an entire function,
L(s,~)
has an infinite number of
0 j Re(s) j i;
L(s,~)
has a factorization
of the type
L(S,~) = eA(~)+B(~)S]T(I - ~)e s/p,
P
where
A(w)
and
B(~)
are constants depending on the automorphic
representation
151
and the product runs over all the zeros of
iii)
L(s,~)
in the critical strip;
The sum
~i~i-i-n
P
extended over all zeros of
L(s,~)
converges for all
q > 0
and diverges for
= 0.
Remark 3.1.1.
The location of the critical strip for
L(s,~)
immediate consequence of the non-vanishing of the Euler product for
and the functional equation.
therefore,
It will be shown in
using the functional equation at
L(O,~)
we also have
3.2.
= E(~)f(~)
§5
that
is an
Re(s) > 1
L(I,~) # 0
and
and let
N(T,w)
s = 0
½
L(I,~),
L(0,~) # 0.
The First yon Man$oldt Formula.
Let
~
be an automorphic representation of
denote the number of zeros of the Euler product
0 < o < 1
and
Itl ~ T.
GL2~A ~)
~(s,w)
in the rectangle:
For the sake of simplicity we assume that
coincide with the ordinate of a zero of
E(s,~).
T
does not
Recall that
r(s,~) = ~-½(S-Xl)r(i~2~1)~-½(s-X2)r(?),
where the infinity type
infinite prime.
depends on the local representation
In particular the trivial zeros of
negative real axis when
mentary series;
{11,12 }
~
~(s,~)
~
at the
are located on the
is a member of the discrete series or the comple-
this lost situation seems to never arise for
GL2~A ~)
and can
152
even be proved for those a u t o m o r p h i c r e p r e s e n t a t i o n s
It
~
b e l o n g s to the principal series then
and the trivial zeros of
C(s,~)
%1
that have no ramification.
and
%2
are not purely real
are located on two rails parallel to the negative
real axis.
For any one of these cases w e have,
E-function,
that
from the n o n - v a n i s h i n g of the
2~N(T,7) = A R arg L(s,Z) + 0(i),
where
R
is the r e c t a n g l e in the
A R arg L(s,~)
perimeter of
term
0(i),
s-plane w i t h v e r t i c e s
is the v a r i a t i o n of the argument of
R
in the counter clockwise sense.
w h i c h is independent of
7,
L(s,7)
3
1
~ ± iT, - ~ ± iT, and
as
s
traverses the
The n e c e s s i t y of adding an error
comes from the p o s s i b i l i t y of
having at m o s t a couple of trivial zeros inside
~(s,~)
R.
The c o n t r i b u t i o n of the left half of the contour is obtained from that of
the right half by using the simple relation, w h i c h is a consequence of the functional equation,
arg L(q + it,q) = arg f(7) ½-s + arg L(I - ~ + it,~) + c,
where
c
is a constant d e p e n d i n g on the root number
8(7)
but not on
s.
We
then w r i t e
~N(T,~) = a L arg L(s,7) - ½ A L arg f(7) ½-s + 0(i),
where
! + iT
2
L
denotes the path going from
1
and then b a c k to ~ + iT.
1
~ - iT
to
3
~ - iT
Now w e clearly h a v e
A L arg f ( ~ ) ½ - s = -2T log f(~)
then
from
3
~-
iT
to
153
and
A L arg ~-½(s-11).z-½(s-12)
also by the complex version
A e arg r(S-ll)r(s-12)
2
2
where
of
Stirling's
formula we have
= (2T - h Ti - 12 ) log ~T _ 2T - ~(!~ + % O1 + 12 ) + 0(T-I),
Ik = %k + iI , k = 1,2.
zN(T,x)
= -2T log z;
We then obtain
the formula
= (2T - h TI - 12 ) log ~T - 2T - ~ ( ½
+ h ~I + 12 )
+ 0(T -I) - 2T log ~ + T log f(~)
+ A L arg ~(s,~)
+ 0(i)
.Tf(~) ½.
= 2T log < ~ )
- 2T + A L arg ~(s,Z)
where
+ 0(Ill
log T),
]I[ = max([%l],[%21).
To obtain the variation
2 auxiliary
of
arg ~(s,~)
along the path
L
we first prove
lemmas.
Lemma
3.2.
We have for large
T,
1
<< log T,
p 1+(T-y) 2
where
strip.
the sum is taken over all the zeros
O = ~ + iy
of
~(s,~)
in the critical
154
Proof.
Logarithmic
differentiation
of the Hadamard
product
formula
F(s,Z)6(s,Z) = eA(z)+B(~)s~(I - ~)e s/P
P
yields
~'
- ~ (s,~)
r'
--F'(s-12) - log z
B(z)
1
i}.
: ½ ~ (S-ll)2 + ½ F
2
- [{s---~p + p '
r v
in this formula we substitute
large
Isl
in the angle
real parts of both sides
the bound
-~ + @ <
~
(s) = log s + 0(Isl-l),
arg s < z + 6,
some positive
is valid for
3
s o = ~ + iT
6,
-½ < o < ~3
and take the
and
t _> 2;
- P
this last inequality
evaluated
at
gives
[ Re{l+
P
1
So-P
where we have used the fact that
~] <
(s,z)
A'
log T,
is bounded
at
So.
Now by Theorem
we have
1
Re ~ = [ Re(0)Ip1-2
O
O
= 0(i),
and also
3
[ ~e{
O
where
for
to get
-Re ~ (s,~) < A log t - [ Re{ _
P
which
valid
P = B + iy.
} =
-
This proves
(~- ~)2+(r-y)2
the lemma.
--
p l+(r-y)
2'
3.1
15,5
The following
Corollary
is a useful corollary
3.3.
to the above lemma.
We have
i)
N(T + l,w) - N(T,w) <
1
ii)
A log T;
< A' log T.
IT- I~i I+(T-Y) 2 --
The next lemma, needed here to compute
in the derivation of the explicit
Lemma 3.4.
~(s,Z)
and
For large positive
3
-½ < ~ < ~
~'
where
P = ~ + iV
Proof.
3
--+it
2
formulas
arg L ~(s,~),
in §4.
t
not equal to the ordinate
i
(s,~) = [ s_--~ + 0(log t),
P
We evaluate
the logarithmic
It-yl > 1
~I 1
p s-p
3.3
that satisfy
It-yl < i.
~(s,~)
of
at
one value from the other to get
p s-p
By Corollary
s = O + it,
derivative
~'(s,~) = 0(log t) + [{ i
the terms w i t h
of a zero of
we have
runs over the zeros of ~(s,~)
and subtract
will also be used
contribute
1
};
at most
I < ~
~3+ i t - p
1
3
~ + it-p
i
0(log t).
p (t-y) 2
we know that the total contribution
p
I~+ it-pl
of the terms in
s
and at
156
with
It-yl < 1
is at most
It should
0(log t);
be observed
this then proves the lemma.
that in Lemma 3.4 the restriction
different from the ordinate of a zero
P = 6 + iy
that
t
be
is clearly not necessary
if
0#6.
To complete
the derivation
of the First yon Mangoldt
A L arg ~(s,~) = S Im ~
formula observe
that
(s,~)ds + 0(I),
(~)
where
term
(~)
0(I)
where
3
~ + iT
is a straight path going from
denotes the variation of
~(s,~)
has no zeros.
arg ~(s,~)
to
~ + iT;
here the error
along the line
Re(s) =
3
Now
/ Im{--i }ds = A arg (s - p)
(~)
s-p
where
a
at most
denotes variation along the path
7;
proof of the following
Theorem 3.5.
representation
%1
and
and this is in absolute value
this remark together with the expression
Lemma 3.4 leads to the estimate
let
(6)
of
%2
for
~
A L arg ~(s,~) = 0(log T).
(s,~)
given in
This completes
the
theorem.
(First von Mangoldt
GL2~A~)
of conductor
be the infinity
type of
Formula).
f(~)
~
and
and put
Let
L(s,~)
~
be an automorphic
its Euler product;
i%i = max(i%ll,i%21).
we have
N(T,z)
= ~2T-
log (Tf(z)½~
.~.
- ~2T+
o(i~i log m),
where
N(T,~) = #{p = 6+i~f: L(p,~)
= 0, 0 < 6 < i, -T < t < T}.
Then
157
3.3.
Explicit Estimates.
The First von Mangoldt Formula given in Theorem (3.5)
depends implicitly
on constants whose values change with varying automorphic representations even if
these have the same conductor and same infinity type.
It appears that these con-
stants can be estimated more explicitly if one imposes various restrictions on the
local components
~
of the automorphic representation
P
7.
We want to consider
here the problem of making explicit the error term in the First von Mangoldt
Formula for an automorphic representation whose local component at the infinite
prime is a member of the holomorphic discrete series.
Let
A
and
k
be positive integers.
let
F (A)
o
~
defined modulo
A
with
subgroup.
~
be an automorphic representation of
Let
~(-i) = (-l)k;
Let
translates of a primitive holomorphic cusp form
Recall that such an automorphic form
f(z)
be a Dirichlet character
be the Hecke congruence
GL2~~ )
of type
generated by the
{~,k,Fo(A)}.
f(z) has a q-expansion
f(z) =
oo
~ a(n)q n,
n=l
with
a(1) = i;
also
f(z)
is an eigenfunction of the Hecke operator
T
P
oo
oo
flTP = n~l a(pn)qn + ~(p)pk-i n=l~a(n)q pn, p~A,
and of the operator
U
P
oo
flUp = n~la(pn)q n, Pl A,
with the corresponding eigenvalues being the
of
~
is given by
a(p).
Recall that the Euler product
158
L(s,~)
= r(s,~)~(s,~)
with
F(s,~) = ~-½(S-%l)F(S-ll)z-½(s-k2)F(s~%----~2),
2
k-i
~I = -~--'
k+l
2 '
~2 = -
and
E(s,~) = p~A i
"p~A
1
l-a (p) p-S
i-~ (p) p-S+~ (p) p-2S'
where
~(p) = a(p)p½(l-k);
L(s,~)
satisfies
the functional equation
L(s,~) = g(~)A½-SL(l-s,~).
Recall that Deligne has proved
([ 7 ], §8.2)
~ ( p ) = kp + Op,
[%p[ =
that
[Op[ :
and Ogg ([25],p. 295)
has proved that when
plA,
can be defined modulo
A/p;
@
l~(p) l = p-½
if
p2~A
l~(p) l = 1
and if
@
if
1
for
~(p) = 0
can be defined modulo
equation and the Euler product imply, as we saw in
L(s,~)
lies in
Theorem 3.6.
if
§2.4,
A/p.
f(z)
N(T,~)
and if
A/p;
The line
L(s,~);
Re(s) = 1
the functional
that the critical strip
0 J Re(s) j I.
Let
~
be an automorphic representation
local component at the infinite prime belongs to the holomorphic
let
p21A
cannot be defined modulo
is the boundary of the region of absolute convergence of
of
p~A,
be a primitive cusp form of type
be as in Theorem (3.5).
We then have
{~,k,Fo(N)}
of GL2~A Q)
whose
discrete series;
associated to
~.
Let
159
2T
TA ½
2T
N(T,7) = -~- log (-~--) - T + (k41--) + 0(k) + 0(log (100Ak2T2)),
where the implied constants are absolute,
meters that characterize
Theorem 3.7.
of zeros
O
Js - ~] i 7
3
of
the automorphic
Let
L(s,~)
7
and
that is they do not depend on the pararepresentations
L(s,7)
7.
be as in Theorem 3.6;
counted with their proper multiplicity
is bounded by
4 -I
(log 5)
log (cAk2),
where
then the number
inside the circle
c = 2(27)
-2
3 4 5 2
~(~) ~(~)
We give only a detailed proof of Theorem 3.7; the proof of Theorem 3.6
follows in outline the same argument used to derive the First von Mangoldt Formula
except that at the crucial point one must replace the bound
A L arg ~(s,7) =0(logT)
by an explicit estimate of a type that will be given below.
Proof of Theorem 3.7.
L(s,7)
inside the circle
formula to the circles
C1
CI:
and
To get an upper bound for the number of zeros of
3
Js - ¼1 j ~
C2: Js - ¼[ ~ "
(~)mi
where
m
is the number of zeros of
maximum modulus of
L(s,7)
The estimates of
we apply the weak form of Jensen's
we thus have
M
L(s,~)
inside the circle
Ogg for the
7,
a(p)
inside the circle
give
M
and
is the
C 2.
and of Deligne for the
and the expression
i
~(s,7)
C1
p~A (I-~(P) P-S) p~A (I-Xpp-s) (I-Opp-s)
%
P
and
o
P
160
5
1¢<¼,~)1
5
-
< ¢(5) 2.
Similarly,
from the identity
co
oo
~(S,~) = p~A(v~O(~(p)p-S)V)
we get for
s = O + it
and
co
p~A (v~O
(%pp-s)v)(= w=0
~ (Opp-S)W),
o > 1
]~(s,~)] < ~(o) 2.
To estimate the maximum modulus of
apply the Phragm~n-Lindel~f
by the lines
s = -½ + it
Theorem ([ 29 ], p. 195)
and
s = 3 + it.
Mellin transform of a primitive cusp form
(3)
where
type
~(s,~)
Since
inside the circle
in the strip
L(s,~)
f(z) of type
is the interval
~,k,Fo(A)}
[A-½,~],
{~,k,Fo(A)}
g(z)
s ÷ s + ½(k-l),
From the above integral representation
S(-½,3)
it follows that
l~(s,~)l j c exp Itl c,
c.
We also have
bounded
the
we have
3
2dy,
representation
~;
this is the formula in [37 ], p. 94.
and satisfies there
for some positive constant
we
is the primitive cusp form of
which is associated to the contragredient
modulo the change of variable
in the strip
and
S(-½,3)
is essentially
L(s,~) = f (y½kf(iy)yS + g(~)A½-Sy½kg(iy)yl-S)y
(~)
(y)
C2
L(s,~)
is regular analytic
161
l~(3+it,~)l < ~(3) 2.
To get an estimate for
l~(-½+it,~)l
~(s,w)
we use the functional equation
E (~) A½-S (2~) 2s-l~ (l-s, ~)
F(l-s- k-l)
F(s + k21)
thus
~(-½+it,~) I _< A(2~)-2~(~)2(t 2 +q-).
k2
The PhragmSn-Lindelof Theorem
strip
S(-½,3)
([ 29], p. 195) leads to the conclusion that in the
we have
k2
[~(s,~)[ j A(2~)-2~(~)4(t2 +-~-).
For
s
inside the circle
maximum modulus of
~(s,~)
C2: Is - ¼1 ~ 7
7
k2
t2
~- +
j 2k 2.
we have
inside the circle
C2
Hence the
is bounded by
M < A(2~)-2~(~)42k 2.
Finally we get
M
5
< cAk 2 '
with
3 4 ~(~)
52
c = 2(2~)-2~(~)
.
We now take the logarithm of both sides of Jensen's inequality to get
162
m < c' log (cAk2),
where
4 -i
c' = (log ~)
.
This completes the proof of Theorem 3.7.
Remark 3.3.1.
inequality to ~(s,~)
C2:
is -
gives
5
- iT 1
To complete the proof of Theorem 3.6
on
~ ~'7'
the
two
circles
the bound for
N(T + 1,7) - N(T,~) <<
el:
~(s,~)
log (cAk2T2),
5
Is - 7 -
with
s
one applies Jensen's
iT I < 13
-i7
and
in the strip
S(-½,3)
then
where now the implied constants are
absolute.
Remark 3.3.2.
The explicit estimates given in Theorems
(3.6)
were obtained under the assumption that the class one local components
and
~
(3.7)
of
P
the automorphic representation
assumption
~
are all members of the principal series;
this
was in fact used in the form of bounds for the coefficients of the
Dirichlet series
~(s,~)
which in turn lead to explicit upper bounds in the
region of absolute convergence.
In order to extend the domain of validity of the
explicit estimates we are lead to introduce a new parameter which measures the
global size of the automorphic representation.
This can actually be accomplished
by two seemingly different methods which we now proceed to sketch.
Method i.
Here we use the fact, established in
Ramanujan conjecture is true on the average:
tation of
GL2~)
,
then there is a constant
if
~
§2.4, that the Petersson-
is an automorphic represen-
II~I] such that for all
x ~ 1
la(n) i << li~IIx,
n<x
where the
a(n)'s are the coefficients of
independent of
~.
To get a numerical value for
to a real analytic cusp form
line of the integral
~(s,~)
f(z),
and the implied constant is
II~II, say when
~
is associated
one studies the growth behavior on the real
163
~o(N)If(~)l2(cI(O)y½(l+O)+ c2(O)y½(l-O))d~,
where
N
is the conductor of
Cl(S)y½(l+s)
+ c2(s)y½(l-s)
~,
Do(N)
a fundamental
is the constant
domain for
Fo(N)
term of the Eisenstein
In the Fourier expansion about the cusp at infinity for the group
already suggested in
of
f(z).
§2.4
one can relate
II~ll to the Petersson
Once the average size of the coefficients
summation yields the estimate for
implicitely
of the local representation
Fo(N).
GL2~A~)
and
T ~ 2,
is known, partial
~
and the infinity type
Method 2.
and
Bounds for
3.6 and 3.7
on
~
equation of
we write the functional
has not been introduced
erature of the subject.
-s
L(s,~):
L(I-s,~);
the dependence of the constants
clearly the global behavior of
where the
can be obtained from known estimates for
L(s,~) = g(~)f(~)
in order to make explicit
<< log (NII~[I~2T),
representation
% = max(l%li,l%2i).
~(s,~)
the F-function and the functional
{%1,%2}
This estimate can then be used to show, following
N(T+I,~) -N(T,~)
implied constant is absolute,
As
inner product
the argument in the proof of Theorem 3.7, that for any automorphic
of
E(z,s)
It should be observed that this estimate
the conductor of
~ .
series
o > 1
where the implied constant is absolute.
already contains
la(n) I
and
L(s,~).
that appear in Theorems
equation in a form that exhibits more
For this we use a bit of notation that
earlier in these notes but which is implicit in the litLet
~
be the subgroup of diagonal matrices
which appears in the Iwasaw decomposition;
if
o E SL20R )
we write
of
SL 2
164
o : (1
° <o))le
0
0
Let ~
be the Lie algebra of
A
e-OH(O)]k(@)"
and ~
its dual.
Let
~
be the nontrivial
^
element of the Weyl group of
Let
A(N)
A
with
be the domain of
~(~) = -I
for any complex number
% C~.
defined by
A(N) : {(~ 0a_l): a2 _> N-½}.
Let
d*a
be a Haar measure on the group
(3)
used in the proof of Theorem 3.7
~.
Then the integral representation
can also be written in the following
alternative form.
Theorem 3.8.
conductor
tation.
f(~)
If
Let
~
be an automorphic representation of
and root number
% E ~
g(~);
let
~
GL2~A ~) of
be its contragredient represen-
then we have
.l+l
L(~--,~) = f
(~(a)e %H(a) + s(~)f(~)½~(%)~(a)e~(%)H(a))d*a,
A(f(~))
where
~
(resp.
~)
is a vector in the representation space of
chosen so that the corresponding primitive cusp form
the isomorphism of
§1.3
~
(resp. ~)
f(z) (resp. g(z)) given by
has its first Fourier coefficient about the cusp at
infinity equal to one.
To recover the functional equation from this integral representation we
observe that
2
is the identity element in the Weyl group and
The key point to observe here is that the functions
defined behavior in the domain
A(f(~)).
~
and
~
g(~)g(~) = i.
have a well
In fact, if we define
II~II = max( max
If(iy)e2~YI, max
Ig(iy)e2~YI) ,
yN~__> 1
y~__>l
165
then the above Theorem or the simpler
expression
<_
where
r(a,x)
(3)
give
+
is the incomplete
(2~)
_
},
f'-function
oo
F(a,x)
= / ta-le-tdt,
x
and
k = ~weight
"0
of
f
There are several well known assymtotic
to explieitely
automorphic
a(z).
it.
representation
Here
therefore
estimate
k = 12
and
if
if
f
f
is holomorphic
is real analytic.
formulas
for
F(a,x)
Let us now give an example.
of
GL2~A ~)
N = 1
f(z) = g(z) = A(z).
corresponding
and also,
~
Let
that can be used
~
to Ramanujan's
be the
modular
is its own contragredient
and
In this case we have
li~ll = max
y>l
: max
y~l
IA(iy)e2~Yl
i~(l
n=l
- e-2~ny) i24
= 1
and so
L (.l+s
__~_~)
To go much further
functional
i _< { F ( ~ , 2 z ) ( 2 z ) - ~ ( 1 2 q < 7 )
+ ~.12-~
y t ~ , z ~ ).t.z.~. . )-½(12-~)~ ~.
than this one must look into the problem of approximate
equations.
form
166
We end this section by proving an inequality that is suggested by the
methods of Stark-Odlysko which have already been quite successful in the problem
of getting explicit estimates for the conductor of an Artin L-function.
Theorem 3.9.
conductor
f(~);
let
f(s) = ~(s,~)~(s) 2
Let
~
L(s,~)
be an automorphic representation of
be its Euler product.
satisfies for
Re(s) > I
GL2~A~)
of
Suppose that the function
f,
-Re ~ (s) ~ O.
Then we have for
Re(s) > 1
Re { ie}
-_< 2R
{i}+_
½ l o g f(~) + Re {½ TF'(s-~-2 +~T"
F''s-%2"(__~)+~'(½s+l)}.
P
where the sum
E
P
runs over all zeros of
f(s) = 0
and
p
and
~
are grouped
together in the sum and are counted according to their multiplicities.
Proof.
(J-P. Serre).
From the Hadamard product formula
L(s,~) = eA(~)+B(~)S~T(I
- ~)e s/p
P
we have by logarithmic differentiation
(s,~) = ½ T
2
+ ½ "~ (
) - log ~ - B(~) - ~
p
and from the functional equation we have
nT
~(0,TT) = B(Tr) = - log f(TT) - B(~) _ ~~ { l !
P
+ l}p
or equivalently
B(~) + B(~) + log f(w) = - ~ [
P
+ %},
+ i}
O
167
where
~ is extended over all zeros of L(s,~). Now we observe that if
O
a zero of L(s,~) then 1 - p is a zero of L(s,~) and similarly if p
zero of
L(s,~)
then
~
is a zero of
L(s,~).
p
is
is a
Hence we can write
Re B(~) + ½ log f(~) = - ½ [@ {i@ + ~}i
and
(4) I R e { _ } - Re ~~' (s,z) = ½ log f(z) - log ~ + ½ Re ~F' ([~%1 ) + ½ R e ~ £' (s-~2),
P
where the
p
and
~
terms are to be grouped together.
of Riemann's zeta function ([ 6 ], p. 88)
1
s-p
Also from the theory
we have
%'
1
- - - ½ log ~ + ½ ~
~ (s) = s-i
(~ + i),
P
where the
O
and
~
terms are to be grouped together.
Adding to (4) twice the
real part of the last equality we get
[ ~e{ s_--i} - Re{ 2~' (s) + 7~' ( s , O }
P
= 2 Re{sll } + ½ log f(~) - 2 log ~ + Re{F'(½s+l) + ½ T (
The positivity of the function
f~
- Re ~ (s)
) + ½-F (
) }"
now gives the desired inequality.
Although we have not used the assumption
f,
is essential~ in fact to verify that - Re ~ (s) ~ 0
Re(s) > i, in practice this
one must employ Dirichlet
series whose domain of absolute convergence lies to the right of the line
Re(s)
=
i.
Example.
(J.-P. Serre)
Let
~
ciated to a holomorphic cusp form of type
be an automorphic representation asso{~,ro(A),k}.
Recall that the infinity
168
type is
3
s = ~
and
%2 = -
0 < o < i.
b o u n d e d by
~
~ = ½
then
over the rationals,
if
k = 2
ml -< 1.7 + ½ log A.
rkE
and
If
E
L(s,~)
m½
L(s,~)
in the
on the real line is
is the m u l t i p l i c i t y of a
is an elliptic curve defined
denotes the rank of the M o r d e l l - W e i l group of
E
then the above estimate together w i t h the Birch and Swinnerton-
Dyer c o n j e c t u r e suggests that
E.
In the inequality of T h e o r e m 3.9 take
Then the number of zeros of
3
.24 + ~ log --(k2A);
over the rationals and
of
k+l
2 "
and let the sum on the left run over the real zeros of
interval
zero at
k-i
2
%1
rkE < 1.7 + ½ log A,
where
A
is the conductor
169
.§4.
Explicit Formulas.
4.1. The Second von Mangoldt Formula.
Our aim here is to prove a generalization,
representations
of
GL2~)
,
in the framework of automorphic
of the famous Second von Mangoldt Formula of prime
number theory
[
p
where
~
P
log p = x - Z ~
x-L- - ~(i)(0)
P p
~(0)
½ log (l-x-2),
<x
runs over the non-trivial
zeros of
((s).
We shall actually obtain a
formula which involves only a finite number of zeros and an explicit error term.
As was already hinted in the introduction
such formulas
principle for automorphic
of other adele groups;
representations
can also be obtained in
we develop a general type of zeta distribution which generalizes
formulas of interest
Throughout
of conductor
f(~)
the Euler product
of the Dirichlet
all known explicit
in number theory.
this section
~
and infinity type
L(s,~)
the 2-dimensional
in fact in §6
of
~
representation
is an automorphic
{~i,%2}.
GL2(~).
of
GL2~A ~)
When convenient we shall write
in the equivalent
of
representation
Let
form
A(n)
L(s,~,r),
where
r
be the n-th coefficient
series
co
-
observe
that
A(n)
--~'(s,~) :
[ A(n)n-S;
n=l
has its support at the prime powers; define for
~(x,~) =
~
x > 0
A(n),
n<x
where the term in the sum with
n = x
is
is to be weighted by
½
when
x
is an
170
integer.
We shall need the following well known result
Lemma 4.1.
Let
6(y)
be the function of
0
6(y) = {%
1
if
if
if
0 <y<
y = 1
y > 1
y
([ 6 ], p. 109).
defined by
1
and let
c+iT
1
s ds
I o (y,T) = 2--~ f
Y "--"
s
c-iT
Then, for
y > O, c > O, T > O,
IIo(Y'T)-6(Y)I
c
< {Y min(l,T-iIlog
cT -I
From this lemma we easily obtain, with
y]-l)
if
if
y # 1
y = I.
c = 1 + (log x) -I,
co
(i)
]V(x,~)-I(x,T)l
< ~ IA(n)](X)Cmin<l,T-111ogTIx
-i
) + eT-11a(x) l,
n=l
where
1
c+iT
~'(s,~)~}ds,
l(x,T) = 2--~ /
{c-iT
and the term
cT-IIA(x)l
is to be included only if
x
To estimate the sum in the right hand side of
contribution
arising from terms with
n _< 3x
and
is an integer.
(i) we consider the
n <_5x.
remains bounded and the resulting sum can be majorized by
T-I~ [ IA(n) ln-~.
n=l
For these
flog xl-i
171
Recall that if the local representation
of
is class one, then, in the notation
P
§1.4, we have
det(l 2 - p-Sr(gp))-i =
~ a(pV)p -vs
v=O
and
oo
--d
l°gds det(12 - p - S r(gp))
=
~ A(pV)p-VS;
v=l
by comparing power series expansions we easily obtain
A(pV) = ~a(p)log p
-a(pV)log p - ~(p)a(pV-2)log p
where
~(p) = det r(gp).
if
if
v = i
v ~ 2,
If we assume, as we have done throughout these notes,
that the central character
~(p)
is unitary, then we obtain the following useful
inequality:
]^(pV) I
(2)
<
--
{la(p) llog p
la(pV) llog p + la(pV-2) llog p
if
if
v = i
v ~ 2.
It can also be verified that this inequality holds even when the prime
the conductor
p
divides
f(~).
From (2) we have
oo
co
iA(n) in-C < ~[a(p) l(log p)p-C + ~ ~ la(pm-2) flog p)p-mC
n=l
p
P m=2
co
<
co
[ la(n) l(log n)n -c + [ log p{p-2C + p-2C ~ la(n) i(log n)n c}
n=l
p
< 3( ~ la(n) l(log n)n-C)(~(log p)p-2C).
n=l
p
n=l
lV2
Using partial summation and the estimates
fa(n) l
for
of
§3.3 we obtain
n<x
co
[ IA(n) In-c << llzll(log x) 2,
n=l
where the implied constant
is absolute.
It remains to estimate
the contribution
to the sum in (i) arising from
the sum
S =
[
3
li(n) Imin(l,T-lllog xl-l).
<
<5
7x_n_Tx
If we knew that the Petersson-Ramanujan
tation
~
Davenport
was true then a relatively
([ ~ ], p. 113)
right hand side of
conjecture
< x >
would lead to the estimate,
constants are absolute.
the Petersson-Ramanujan
3
~x<
x
+ min(l, T--~-~x>},
the
inequality
x
to the nearest integer and where the implied
To obtain an estimate for
S
without
the assumption of
conjecture we are forced to take a different route.
n < x;
We first divide the sum
satisfies
for the whole sum in the
(i),
is the distance from
treat the case
represen-
simple argument similar to that given by
<< II~II (log x){ x l ° g x
T
where
for the automorphic
S
the
other
case
into two sums
(a)
or
(b)
can be
Sa
and
below:
a)
T-i llog--Xni-i < N < I,
b)
T-iIlog ~I -I ~ ~,
treated
sb
by a similar
We
argument.
depending on whether
n
173
where
N
is a fixed number
to be selected later.
In case
n
satisfies
(a)
we
have
Sa
-n < x
If we replace
[A(n) l
by the inequality
in (2) and use the estimate of
§3.3 we
obtain
s a << N ll~IIx log x.
The sum
sb
runs over those
n < x
for which
!
> log (~)
Tn -n
or equivalently
n _> x exp {- T!}.
For these
that
n,
(~T) -I
the sum
sh
is bounded,
has at most
1
x(l - exp {- T~})
then the number of terms in
If we now apply the Cauchy-Schwarz
c(n)[A(n)[
3
-~x<n< x
inequality
terms, and if we assume
sb
is at most
to
sb
1
<< --'TN
we get
<< ( ~ c(n)2)½( ~ A(n)2) ½,
n< x
n<x
---
1
where
A(n)
c(n)
is
i
if
by the inequality
x exp{- ~ }
(2)
< n < x
and
0
and use the estimates
s b << (xN-iT-l)½(log
otherwise.
of
§2.4
x) llxllx½.
and
Again we replace
3.3
to obtain
174
We thus have
S = S a + sb
<< II~llx(log x){n
We now select
N
so that
~ = N-½T -½,
~-½T-½}.
+
that is
1
n = T3
•
with this choice we have
i
s << 11~11x(1og x ) r - Z
A similar argument works for the sum in the range
the above estimates and using the fact that
x < n
< 5
~x.
Putting together
IA(x) l << II~]l x log x,
we now have
i
l~(x,7) - I(T,x) I << 11711 x (log x) 2T -I + [1711 x (log x)T - ~
<<
where the implied constant
To evaluate
satisfying
the
the condition
I1~11
x
1
(log x)2T -~,
is absolute.
integral
l(x,T)
that for any
u
we take a positive real number
in the set
Z(~ ) = {%1- 2n, % 2 - 2n:
we have
i
IU + Re(u) l _> ~.
Let
Lo
U
n • ~,
n ~ 0},
be the contour consisting of the segments
175
L : -U < O < c, t = T
: - U < (7 < c, t = -T
where
c = i + (log x) -I
of the zeros of
L':
O = -U,
0 < t < T
L':
(7 = -U,
-T < t < 0
L :
c
O = c,
-T < t < T,
and
T
is s u i t a b l y chosen so as to a v o i d h i t t i n g any
~(s,~).
Cauchy's r e s i d u e theorem yields
l
(3)
v
s
o
¢(2)
(log x + ½ ¢(i)(0,7)
w h e r e the first term, to be i n c l u d e d only w h e n
_
p
xp
~.
v-
{ -i-,
s = 0
is a zero of
~(s,~),
arises from the obvious Laurent expansion
~'
the sum
sum
~
1
~(2)(0,~)
(s,~) = -- + ½
+ ~ A(n)sn;
s
~(i)(0,~ ) n= I
E runs over the n o n t r i v i a l zeros of
~(s,~)
P
runs over those trivial zeros
% of ~(s,~)
consider now the integral in (3)
integral a l o n g
t = T
L
Z(~ ) n L o
§3.3
T - i < T < T + i
is
<< i,
where
= [1~11 log (f(~)(T + I~ll + I~2[>>"
# 0.
We
We divide the
- U < o < -½, t = T.
that the n u m b e r of zeros
and the
and
the first c o n s i s t i n g of the segment
and the second consisting of the segment
the critical strip w i t h
in
llmpl ~ T
along each individual path.
into two parts,
the first piece w e recall from
with
-½<
To
O = ~ + iT
O<
treat
in
c,
178
Therefore,
by slightly
for any zero with
increasing
T
T- 1 < y < T + 1
this were not possible,
w o u l d be m u c h larger
if necessary,
the number of zeros
for
Let us recall
~
(s,z) for
from
§3.3
s
P = B + iy
of
~.
with
E,
(s,~) = 0(log(It I +
Now we observe
and
2 + it
llll +
IX21) +
that
1
Is-{3
and summation by parts,
these terms contribute
w e have
and subtract
t
~'
i
derivative
}
of the
§3.3
<<
II~II;
IY-tl ~ 1
with
w e have
2-0
I(s-p)(2+it-p)
<< £.
3
<
-IY-tl 2 '
N(T+I,~)
Similarly
- N(T-I,~)
<< ~,
for those terms with
and these are in number at most
<< ~.
~ (s,O =
~,
i
s-p
+
shows
Iy-tl < 1
and
-i<o<
0(~),
P
where
the sum
constant
Z runs over those
p for which
It-YI < i,
P
is absolute. From this representation we obtain
that
In conclusion we
(not coinciding with the ordinate of a zero)
~'
(4)
-P
to get
for the logarithmic
by
1
I 2+it-o
at most
further w e
t = T.
IF (2+it,~) I + ~{slo
- - 2+it-p
o
using the fact that
12 + i t - p] > 1
have that for large
E
P
If
_ log w - B(~) - [{~_in + ~};
1
I~'(2+it,~)I
also for the terms in the sum
A.
the expression
w h e r e w e have used well known bounds
F-function.
constant
Before proceeding
p
it at s = ~ + it
IT-y1 > A~I
T- 1 < y < T + 1
along the chosen rail
!' (s,~) : ½ yF' (s~ ~i) + ½ ~r'(s-~2)
7--
we evaluate
that
and with an absolute positive
than a constant multiple
must find an estimate
we may assume
and the implied
2,
177
l~'(q+iT,~)l <<
~
,% + 0(%)
I~ l<l
<< %2,
uniformly for
-½ < ~ < c,
where the implied constants are absolute.
the contribution to the integral in
(3)
Therefore
along this part of the line
L
is at
most
c
<<
s
~2 f ]Xldc~
-½
2
-c
<< % T -lj x°do
-½
<< ~2T-Ix.
A similar argument can be given for the corresponding part of the line
estimate the contribution along the remainder of
l~'<s,~)l << ll~IIlog
which is valid for any
s = o+it
with
L
To
we make use of
(f(~)(21s I +
l~ll + Im21)>,
O __< -½
]s+v] _> !4,
and
[.
for
v E Z(~ );
to prove this one again uses
~(s,v) = O(log(21sl +
l~ll+ 1121)) + ]~'(2+it,~)l+Z{~p
2+1_0],
P
the whole sum
similarly for
Z is now estimated as before using the fact that Re(s) J -½;
P
~ (2+it,~). The term log(21s I + 1%11 + I%21) results from
estimating the logarithmic derivative of
for which
Is+vl ~
i,
F(s-%I)F(s-%2)
2
2
at points
s = o+it
v ~ Z(~ ).
The contribution of the integral along the remainder of
L
and
L
is
178
not larger than
-½
<<
T-l/
-U
<< £x
-½ -i
T .
We therefore get
v
s
1 / {-$ (s,~) x-- }ds << xg~2T-I.
2~i L+L %
s
The integral along the path
L' + L'
<< u-lll~llXog<f(~)(2u +
is at most
lhl +
T -U
Ix21))/ x
dt
-T
<< TU-Ix-UII~IIIog(f(~)(2U +
If we now assume, as we may do, that
as
U + ~.
I%1] +
I%21)).
x > I, then the last expression goes to zero
Putting together the above estimates we get for
P(x,~) = -(log x + ½ ~(2)(0'~)) ~
x0
C(1)(O, ~)
IYT!T P
x > 1
1
x__l + R(x,T),
e z(~)
x__
where
1
R(x,T) << x(ll~lllog{xf(~)(T + I%ii + I%21)})2T 3,
Z(~ )x = Z(~ ) - {0},
and the implied constants are absolute.
Since the product
179
p(S-Xl)p(s-X2)~(s,w)
2
is entire and
that
~(s,~)
Re(>`I) ~ 1
2
is free of zeros to the right of Re(s) = 1
and
Re(% 2) ~ i.
discrete series we have
Re(h i ) J 0.
In fact, if
If
~
~
it follows
belongs to the Holomorphic
is a principal series representation
then Rankin's trick applied to the Euler product
L(s,~)
as in
§2.4
using the positivity of the coefficients of the Dirichlet series
Re(>`i) j ½
always.
~(s,~X~),
Langlands has shown that the stronger result
would follow from the analytic properties of the Euler products
using the simple identity, valid for
~
shows,
that
Re(>`i) = 0
L(s,~,p).
Now,
not an integer,
~, x %-2n = x___
>` _ x___
% log ( l _ x - 2 )
n=O%--~-n--n X
2
+ X%n~l (~n'X-2n'%
we have
>`~
where
Z
x_
>, << x > ` ( O ) { l o g
(1 + x - 2 ) + max (j>`ll +
I~1 I-l, 1>'21 + 1~2 l-l)},
z(~) x
%(o) = max (Re(>`l),Re(>`2)). Observe that if
complementary series then
~
does not belong to the
%(o) < 0.
We now put together our main result in the following statement.
Theorem 4.2.
representation of
(Second yon Mangoldt Formula).
GL2~~ )
llg]l be the norm of
~(x,~)
g
of conductor
f(~)
introduced in §3.3.
= - (log x + ½ ¢(2)(0,~))
¢(1)(0,~)
where the sum
E
P
Let
~
be an automorphic
had infinity type
If
{%1,>`2}; let
x > 1 and T > i, then
xp + s(~)
_ ~ "6-
+ R(x,T),
p
runs over the nontrivial zeros of
~(s,~)
with
llm(p) l ~ T,
180
_i
R(x,T) << ~(ll~rlllog {xf(~)(m
I>11
+
+
IX21)))2T ~-,
and
o~
xkl_2n
( oo) : n:O Oh--7
xk2-2n
+
<< xl(O){log(l+x -2) +max(Ill{ + IliI-l, II2 ] + ix21-1)},
I(o) = max(Re(Xl) , Re(X2));
furthermore all the implied constants are absolute.
(2)
Remark 4.1.1.
be included only if
the terms with
4.2.
As already observed, the term
[(s,~)
~i = 2n
and
has a zero at s = 0;
X2 = 2m
(log x + ½ " [
(0,7))
~(i)(0,~)
the sum
S(~ )
is to
does not contain
if they exist.
Examples of Explicit Formulas.
If
~
is an automorphic representation of
2-dimensional complex representation of
GL2(¢),
GL2~),r
X
is the
its character and
[(s,~) = ~ d e t ( l 2 - p-Sr(g_))-i
P
P
is the associated zeta function, then we have for a prime power
A(p n) = x(g~)log P.
If we now let
x
be fixed and let
T ÷ oo in the formula for
~(x,Z),
obtain the explicit formula
[(2)(O'~r)) - ~ -x p + S(~ ),
T(x,~) = - (log x + ½ ~(i)(0,~)
p
then we
181
where the sum
E
is to be understood as
P
Now let
p~A
E
lim
T+~
[
xP/p.
1Y[ j r
be an elliptic curve defined over
~
of conductor
A.
For
let
Card EOFp) = p + I - a(p),
where
a(p) = Tr(~p)
is the trace of Frobenius acting on
count the number of points of
E
%-adic cohomology,
defined over the p-element field
~
.
Suppose
P
that the Hasse-Weil zeta function
L(s,B) = (2~)-SF(s)~(s,E)
=
(2~)-SF(s) ffA ( I - a ( p ) p - S ) - I v (l-a(p)p-S + pl-2s)-i
p~A
is actually the Euler product
conjecture ([ 40 ], p. 156)
L(s+%~2,~) of an automorphic representation as Weil~
suggests.
Suppose also that the Birch and Swinnerton-
Dyer conjecture is true, that is to say, the multiplicity of the zero of
at
s = 1
is the rank of the Mordell-Weil group
E(Q).
L(s,E)
Then, under these restric-
tive assumptions, we can specialize the explicit formula of Theorem 4.1 to yield
Tr(~)v log p = -x rank E(Q) - [
p#l
p~< x
where the sum
E
P
critical strip
runs over the non-trivial zeros
f(~)
~(i) (O,E)
p
of
~(s,E)
inside the
3
½ < Re(s) < ~.
Finally if
conductors
xO
~(2) (O,E)
~ - ½"
- log (x-l),
and
~
and
f(~')
~'
are automorphic representations of
and if
GL2~A ~)
of
182
L(s,~ x z ' , r @ r )
4
= ~
_ ½ ( s _ % i ) F ( ~ ) ~4( s , ~ _
z
i=l
is the hybrid Euler product associated to them in
x ~ ' ,r@r)
§2.3, then one proves by a more
elaborate analysis an explicit formula of the same type as in Theorem 4.2.
More
precisely, if
(s,~Xz',r®r)
x(g~)x(g~n)(log p)p-nS,
=
P
then, for
x > i
and
n
T > i, we have
xp
Z X(gp)X(gpn)l°g P = x6(~,~')- Z m ( p ) ' ~ + S(~ , "iT') + R(x,T),
pn<_x
P
where the sum
~(s,~X~',rOr)
E runs over all the nontrivial zeros (and poles) p of
P
with
IImpl j T; each counted with its proper multiplicity
4
m(p);
oo x%i_2n
S(~°°'~') = i:l~ n~O= Xi-2n '
and
_i
~(x,T) << ll~lloxT](log{xf(~)0(T
117110 max(ll~ll ' II~II)2'
f(~)O = max(f(~),f(~'))
=
Here
6(~,~') = i
if
~'
+
and
is the contragredient of
~
%0)})2,
%0 =
l~ll
+
Ix21 I~31 I~41"
+
and zero otherwise.
+
The
key point in the derivation is the explicit estimate
N(T +
1 , ~ x ~ ' ) - N ( T , ~ x ~ ' ) << ll~llolog(f(~)o(T + %0) )
for the number of zeros (and poles)
critical strip with
T < y < T + i.
p = ~ + iy
of
L(s,~X~',r@r)
inside the
183
§5.
5.1.
Zero Free Regions for
A Hadamard-Landan
L(s,w).
Type Inequality.
Our purpose in this section is to derive zero free regions inside the
critical strip
0 ~ Re(s) j 1
for the Euler products
L(s,~).
The possibility
of obtaining such zero free regions was suggested by Rankin's work
[ 30 ] where
it is shown that the Euler product
L(s-y,~)11
= (2~)_SF(s)~(l-T(p)p
-s+pll-2s)-l,
P
associated with Ramanujan's
function
%(n),
13
Re(s) =--~
does not vanish on the line
An attentive reading of Rankin's article also suggests how his method, with a few
modifications,
may give the non-vanishing
to any automorphic
representation
of
of the Euler products
GL2~)
on the line
result that we prove below
which is comparable
the only
L(I,z) # 0,
simple nature.
L(s,~)
for Dirichlet L-functions.
convinced that the methods also work for adele groups other than
convolution method is available,
but
The main
(see Theorem 5.1) gives a zero free region for
to those that are possible
an analogue of Rankin's
associated
Re(s) = i;
point that may not be clear in Rankin's method is the proof of
this only requires an extra argument of a relatively
L(s, )
We are
GL2~),
in particular
where
for
GLn~)-
Throughout
tation of
GL2~ ~ )
Euler product
this section we assume that
~
of conductor
type
L(s,~).
given in Rankin
We now proceed
~
where
~
represen-
and associated
similar to that
of the logarithmic
We will treat only the case of an automorphic
which is formally real,
L(s,~) = L(s,~),
{%1,%2}
to derive an inequality,
[ 30 ], which will give the positivity
of certain Euler products.
sentation
f(~), infinity
is an automorphic
that is to say a representation
is the contragredient
of
~.
~This has now been proved by Jacquet and Shalika in [ 15 ].
derivative
repre-
satisfying
184
W i t h o u t loss of g e n e r a l i t y w e may assume that all the local components
of
~
are class one representations,
of
~
has the form
and in p a r t i c u l a r that the zeta f u n c t i o n
~(s,~) = ~ ( l - a ( p ) p -s + @(p)p-2S)-i
P
= ~(l_%(p)p-S)-l(l-~(p)p-S) -I.
P
This is p o s s i b l e b e c a u s e
~
contains at most a finite number of r e p r e s e n t a t i o n s
that are not class one and the local factors a s s o c i a t e d to any one of these can
have at w o r s t only poles.
Let
%(p) = exp i (p), n(P) = exp -i@'(p),
where
8(p) = ~(p) + iB(p)
real numbers.
~(s,~®~)
and
@'(p) = ~'(p) + iB(p)
and
~,e'
and
B
are
W e also need to consider the f u n c t i o n
= ~(2s)~(s,~X~)
=~(i-
% ( p ) ~ ( p ) p - S ) - l ( l - % ( p ) ~ ( p ) p - S ) - l ( l - ~(p)n(p)pS)-l(l-~(p)~(p)pS) -I.
P
F r o m the results of
Re(s) > ½
~(s,n®~).
except at
§2
w e know that
s = i
~(s,~ × ~ )
w h e r e it has a simple pole;
W e use these functions
to show that
enough to show this for the m o d i f i e d zeta
~(i,~) = 0.
s = i
the same holds
L(I,z) # 0.
~(s,~).
Now Rankin's c o n v o l u t i o n method
has a double pole at
is a n a l y t i c to the right of
Suppose
(§2.)
true for
Clearly it is
~(i,~) = 0, then also
shows that
~(s)~(s,~®~)
and h e n c e the f u n c t i o n
H(s,~) = ~ ( s ) ~ ( s , ~ @ ~ ) ~ ( s , ~ ) ~ ( s , ~ )
is r e g u l a r in a n e i g h b o r h o o d of
s = i.
O b s e r v e that
~(s,~)
can h a v e at w o r s t
185
a simple zero at
s = i;
s = 1
for otherwise
but this contradicts
the function
the fact that for
H(s,z)
Re(s)
w o u l d have a zero at
> 1
co
H(S,~)
=
~ A(n)n -s,
n=l
A(n) >__ 0
as can easily be seen from
-hE
log H(s,~)
[
=
[ ]i+~<p)n+~(P)~]2 P
p~f(z)
convergence
coefficients
imply that the real point on the line of absolute
ticular
Clearly
H(I,~)
s = i.
# 0
H(s,~)
H(C,~)
# 0
for
and log H(s)
Recall Landau's
Theorem:
for
n
The absolute
singularity.
of
n= 1
Re(s)
> 1
and the positivity
o > abscissa
is regular
convergence
of convergence
is a
and in par-
in a small n e i g h b o r h o o d
if the Dirichlet
of its
of the point
series
co
F(s) =
where
c(n) ~ O,
gO < c ~ c I
and
F(s)
~ c(n)n -s
n=l
is analytic
of the real axis,
for
C > C1
in a n e i g h b o r h o o d
of the segment
then
co
F(s) =
Now let
g0
Landau's
Theorem,
~ c(n)n -s
n=l
be the first real zero of
the Dirichlet
for
H(s,~),
C > C0"
if it exists;
series
co
H(s,~)
in particular
=
Z A(n) n-s,
n=l
holds
for
O > O0;
-~ < o0 ~ i.
By
186
log IH(o,~) I = Re log H(O,~)
= log H(g,~)
=
for
g > g0"
But then
hence
IH(~)I ~ 1
H(O0,~)
= 0;
g0
H(s,Z)
must vanish to the left of
for
oo
~ a(n)n -g _> 0
n=l
o > gO
does not exist.
contrary
to the fact that
But this contradicts
Re(s) = 1
the fact that
at an infinite number of points on
the negative real axis in order to offset the poles present in the
appears in the functional
that
L(I,~) # 0.
automorphic
equation relating
To get the non-vanishing
representation
~ ®wit,
where
H(s)
of
with
H(l-s).
L(l+it,~),
~it = ~0
and
P
character
r-factor that
This then proves
we replace
~
To derive a zero free region we use the well known inequality
(i)
to obtain for
{3~H ( g , ~ ) + 4 ~
s = ~ + it, ~ > 1
(g+it,z) +~'(o+2it,z)}
~ II+%(P)n+~(P)nI2(Iog PlP
i3+4 cos (tlog p) +cos(2t log p)}
p~f(g) n=l
> O.
Let for
t > 0
e = e(t) = Ii~II2 log(f(~)(t+
and recall from
§3
the representation
1~ll+
by the
is the grossenP
~ (x) = Ixl it
P
P "
3 + 4 cos e + cos 28 > 0
~
I~21)),
187
(2)
-
where we now assume that
to representations
~
(s,IT) = - Z ~
P
~(s,~)
+ 0(%),
does not contain the local factors corresponding
which are not class one.
To see why this is so observe
P
that for
d > 1
l~slog ]~ (l-a(p)p-Sl
plf(~)
<
--
<<
[
p]f(~)
(io$ p)[a(p)Ip -O
l-]a(p)]p
--O
log f(~).
Similarly we have
{'
(3)
-~ (s,{)
For
{(s),
1
= -~ s - ~ + O(Z).
P
modified at the local factors corresponding
to primes
plf(z),
we use the well known result
(4)
_ !'
1
[
i
+ O(log
~ (s) = s---']-s-----p
P
f(~)(l
We did not derive in §3 a result for
there can be used without much difficulty
(5)
-
We thus obtain,
~'
(s,~)
1
s-i
+ I~))
"
6(2s)~(s,ZX{)
to prove
1
[ --
P s-p
+ o(~).
on taking the real parts in (2),
(3), (4) and (5), for
Reo > 1
(6)
but the arguments
-Re ~H' (s,~) <__2 R e { l } _ ~
R e { l } + ci%,
O
188
where
the sum
l
p
runs over the zeros
IY - tl < I; observe
zeros as we want.
positive
of
that since
sponding
Re(s-p)
> 0,
Here and in the following
and absolute.
E(s,~).
P = B + iy
We now choose
By retaining
to the zero
t
of
we may retain
the constants
of
L(s,~)
and
(6)
L(s,~)
with
3
ReD > ~
and
in the sum as few
Cl,C2,...,etc.
to be the ordinate
in the right hand side of
B + iy
H(s,~)
¥
of a zero
only the terms
we obtain
are
B + iy
corre-
in the region
t _> c0~(1) -I
(7)
-Re ~
We omit all zeros and poles
and
in
(o + it,n) _< -
H' (o + 2it,~)
+ c2Z.
in
(6)
and obtain
for
t _> c0%(i~i
~ > 1
H t
(8)
Also from
-Re ~
(o + 2it,~) ~ c3~.
-Re ~
(0,7) <_
(6) we have
(9)
The three estimates
inequality
(7),
(8) and
(9)
when substituted
(i) give
6
0 _< o-i
Take
+ c4~.
o = 1 + 6~ -I,
where
6
8
o-B + c5~"
is a positive
< 1 -
( 86
constant.
Then
_ 6)%-i
6+6c 5
and if
6
is suitably
chosen with respect
to
c6
~3 < 1 - ~ - ,
c5
we have
into the basic
189
where
c6
is an absolute
For
conjugate
then
0
<
t
small but different
from zero we must exploit
the presence
zeros above and b e l o w the real axis in a small n e i g h b o r h o o d
t < c0%(i)-i
also a zero of
~(s,~).
constant.
and let
~(s,~);
similarly
Again w e have for
- Re ~
(10)
p = $ + iv, y = t,
~ = $ - iy
of
i.
be a zero of ~(s,~);
is a zero of
~(s,~)
of
it is
and of
o > i
(O,~) - c7~ < 2 - 2 Re{
-- o-1
} - 2 Re{
}
d-1
2
<
-- o-I
provided
the following
condition
2
o-$
is satisfied:
(*)
t < o - B.
Similarly we have
(Ii)
- Re ~
(o+it,z)
- c8% <_ 2 R e { ~ } - 2
Re{
}-2
2
2
2(o-$)
d-1
o-6
(o_~)2+4t2
2
<
•
-- o-1
_
Re{
1
121
5 o-6
- - , - -
H v
(12)
- Re ~
(o+2it,~)
- c9% i 2 R e { ~ i}
- 2 R e { ~i }
< 2
-- o-i
2(o-B)
(o_B)2+t2
2
<
--o-i
61
5o-6"
2(o-B)
(o_B)2+9t2
Let
- 2 R e { ~ i}
190
The estimates
(i0),
(ii) and
(12)
and the basic inequality
(i) give
16.8
~--B ! ~
If w e take
o = 1 + ~-i,
+ Cl0 ~"
we get
< i - 2i--7'
where
O
6
-i
is chosen p o s i t i v e and
<_ .32
Now o b s e r v e that for the chosen
Cl0
we have
o - B _> ~ ( i . 0 4 ) .
Hence the condition
(*)
6
is a u t o m a t i c a l l y s a t i s f i e d if
It remains to c o n s i d e r the s i t u a t i o n w h e n
t < --
t = 0.
H e r e w e can no longer
use conjugate zeros but w e can appeal to the earlier a r g u m e n t using only the function
H(s,Z).
zero.
Let
~
We w a n t to show that near
and
~
fundamental inequality
be two zeros of
s = 1
there can b e at m o s t a real simple
H(s,Z)
near
s = i;
then w e h a v e by the
(i) and (6)
H ~
0 i - Re~ (o,~)
2
2
i o-i
o-~
2
o-~ + Cli ~ l j~"
or e q u i v a l e n t l y
o-~ +
If
O
is taken to be
i + 60%(i)-i ,
60
this last i n e q u a l i t y shows that
<--
+ c12%(i) "
for a s u f f i c i e n t l y small p o s i t i v e constant
B
and
~
cannot b o t h be greater than
191
1 - 6'%(i) -I
regions for
for a suitable positive
L(s,%)
when
% # ~
6'.
The problem of getting zero free
is somewhat simpler and follows more closely
the classical derivation of a similar result for Dirichlet L-functions
with
X
L(s, X)
a complex valued character.
We collect our results in the following theorem.
Theorem 5.1.
There is an effectively computable absolute constant
with the following property.
of conductor
f(~),
the Euler product
If
infinity type
L(s,~)
~
is an automorphic representation of
{11,12 }
and norm
l[zll and if
c13
GL2~~ )
z # ~,
then
has no zero in the region defined by
O ~ 1 - c13%(t) -I
if
Itl ~ 1
~ i - c13%(i)-i
if
Itl ! i,
and
where
%(t) = ll~H2(log f(~)(Itl + I%11 + I%21)).
If
~ = ~,
the only possible zero of
L(s,~)
in this region is a single
simple
zero.
Remark 5.1.1.
If we assume the Petersson-Ramanujan conjecture for the
automorphic representation
~
then we can use the somewhat simpler inequality
12 Re ~ (O+it,~)l i -2 ~ (O) - Re ~ (O+2it) - ½ R e ~ ( O , ~ x ~ >
which is obtained from the trigonometric identity
192
14
Remark 5.1.2.
contragredient,
products
cos A cos
If
~
BI
< 2 + cos 2A + cos 2B.
is an automorphic
then the above estimates
L(s,~X~)
representation
and
lead to zero free regions
similar to those obtained in Theorem 5.1.
~
its
for the Euler
For this one needs
to consider only the function
g(2s)~(s,g xZ)
Z lZ(p) n
= exp{
p#f(z)
If
~
and
7'
are
one can show that
two arbitrary automorphic
L(l+it,~xT')
# 0
+ q(p)nl2 P -sn/n}-
n=l
representations
for any real
t.
of
and uses the information about L(s,~ x3)
5.2.
x~')L(s,~' x~')
just mentioned.
Prime Number Theorems.
If
~
is an automorphic
representation
of
GL2~ ~ )
co
~(S,g)
-
=
[ A(n)n -s,
n=l
then the explicit
formula of
T(x,O
§4
gives,
=
for
x > 1
Z A(n)
n<x
= -
with
[
xp
-
+ s(~
then
Here we simply consider
the Euler product
H(s) = L(s,n X ~ ) L ( s , g X ~ ' ) L ( s , ~
GL2~ ~ )
) + R(x,T)
and
193
i
R(x,T) <<
xr-~(ll=iI log
(xf(~)(T+
I~11+
I~21))) 2
and
Is(~)l
<< x % ( ° ) { l o g
(l+x-2) + max (l~ll
+
IxlJ-l, lx21 + l~2 [-1)}"
The zero free region given in Theorem 5.1 now yields
[xP[ << x exp{- cl3£(T)l°gx}
where
£(T) = ll~ll2(log(f(~)(T+l~iI + l~21)));
by the estimate of
§3
and Theorem 5.1
i
*
i << £(T) 2
IY <T ~
where we have excluded the exceptional zero
companion
i-8.
if it exists as well as its
From this there results the estimate
x6
(12)
B
~(x,~) + ~ - +
xl-~_l
i-6
- s(~)
Cl31og
£(T)
x}
as a function of
x
<< x(ll~ll21°g(xf(w)(T+ 1111 + 112 l)))2(exp{-
To select a suitable value for
T
impose some condition on the size of the conductor
f(w)
we suppose that
h
f(~) < exp{cll~ll-l(log"
"
x) ~},
-1
" + T 5).
we must first
in relation to
x.
If
194
where
c
is any positive constant and choose
T + Ixll + IX21 : exp{- cll~ll-l(log x)½},
then all the terms on the right hand side of
are of the order of magnitude
<< xll~H2exp{-c, ll~II-l<log x)½},
(i3)
where
z.
(12)
c'
is a constant depending on
c
but not on the automorphic representation
We also have
xl-S_l
i_~
= x
for some
o
between
0
and
I-B
log x
and the last expression is less than the
expression in the right hand side of
(13).
We have thus proved the following
result.
Theorem 5.2.
conductor
f(~),
Let
~
be an automorphic representation
infinity type
{Ii,~2}
and norm
II~ll. Let
of
c
GL2~ % )
of
be any positive
constant and suppose
f(z) ! exp{clIzll-l(l°g x)½}"
Then
6
A(n) = _ ~
+ 0(xl(O)max{Ill I + IIi I-l,ll21 + 1121-1})
n<x
+ O(xll~II2exp{-c, ll~ll-l(log x)½}),
where
c
I(o) = max(Re(ll),Re(12));
c'
is a positive constant depending only on
and all other implied constants are absolute.
Here
6
is the real zero of the
195
Euler product
L(s,~),
if it exists, satisfying
6 >l-6(ll~l121og(f(~)(l+llll+IX21))) -I.
The elementary estimate
A(n) = ~ X(gp) log p +
n<__x
p <_x
([l~ll~=(logx)2),
which is deduced from the inequality given in §4.1 leads to the following result.
Theorem 5.3.
Assumptions and notation as in Theorem 5.2.
If
- ~(s,~) = ~ X(gp)(l°g p)p-nS,
pn
then
x B + 0(x%(O)max{[%ll + II1 l-1 ,1%2] + I%21-1})
X(gp) log p = - ~
pjx
+ 0([l~II2x exp{-c'll~ll-l(log x)½}).
Remark 5.2.1.
~A
If
A(2),
the Ramanujan modular form
ii
max(Re(~l),Re(~2) ) = - ~-
is the automorphic representation associated to
then the conductor is
1
and
%(o) =
and hence Theorem 5.3 gives
ii
p 2 T(p)log p << x exp{-c' (log x)½}.
p<_x
Remark 5.2.2.
let
Let
f(~)0 = max(f(~),f(~'))
max(%(o),%'(o)).
following estimate
~
and
and
~'
be automorphic representations of
I1~1[ 0 = max(ll~ll , liT,I])
GL2~);
and let ~0 =
The same argument leading to the proof of Theorem 5.2
gives the
1 g6
(14)
X(gp)X(g~)~
log p = x~(~,~')
+ 0(x%0-% 0) + O(x exp{-c'(log
x)½})
pix
where
6(~,~')
~en
= 1
if
~ = ~'
Ramanujan modular
7'
is the contragredient
is the automorphic
to a-results
~
and zero otherwise.
representation
form, Rankin ([ 31 ], p. 247)
version of Theorem 5.3 without
of
associated
to the
had obtained a somewhat weaker
the error term and gave an interesting
for the size of the eigenvalues
of Hecke operators.
application
Other applica-
tions of the hybrid formula will be given in future publications.
5.3.
The Problem of Exceptional
Zeros.
Our program of eventually
automorphic
tation
representation
itself necessitates
utilizing
to obtain information
about the automophic
that we have explicit
information
regions inside the critical strip for
tion about the possible exceptional
an exceptional
the explicit formulas associated
L(s,~)
zero.
well known in number theory.
represen-
about the zero free
and in particular
further informa-
The proof of the non-existence
zero will have many far reaching consequences
to an
of such
some of which are
At present there seems to be three approaches
to
these problems:
(I)
To use the Saito-Shintani
theory of liftings [22]
tations to obtain an analogue of the Brauer-Siegal
(II)
To exploit more closely the implications
to non-abelian
class field theory.
morphic representations
representations
of the Weil group
WL
and n-dimensional
effective Brauer-Siegel
estimates.
semi-simple
that the problem of exceptional
zeros arises only for a very special subclass of representations.
tion one finds a natural interpretation
theory [ 16 ]
the relation between auto-
GLn~ ~ )
suggests
represen-
Theorem.
of Jacquet-Langlands
More precisely,
of the adele group
of automorphic
for some of the results of
For example if
f
In this direcStark [ 39 ] on
is a cubic polynomial
and
197
its splitting field
and if
~K(S)
quotient
tation of
K
over
is the Dedekind
~K(S)/~(s)
GL2~A~).
~
has Galois group the diahedral
zeta function of a cubic subfield of
is the zeta function
Now
~(s,~)
is a quadratic extension of
~(s,~)
Q,
computable
representations
associated
K,
of an automorphic
has an effectively
because it really comes from an automorphic
L
group of order
on
G
then the
represen-
zero free region
GLI~AL),
to a grossencharacter
where
of order
3.
In this situation one can apply the classical arguments of Hadamard to get effective estimates.
(III)
A far more interesting
consists in considering
suggestion has been given by Serre.
the Euler product
~(s)6~(s,~,r)7~(s,~,r2)4~(s,~,r3
where
r
square and
is the 2 dimensional
r3
its symmetric
for the exceptional
representation
cube.
~ of
of
GL2(~),
zero can be obtained if the zeta
GL2~)
),
r2
its symmetric
The claim is that an effective
approach is not entirely unrelated to
representation
His idea
to other
(I),
~(s,~,r 3)
but involves
GLn'S.
upper bound
is entire.
This
lifting the automorphic
198
§6.
Zeta Distributions.
6.1.
Introduction.
In this section we let
defined over
~
and let
~
G
be a connected reductive algebraic group
be a representation of the adele group
occurs in the space of cusp forms and let
structed according to Langlands'
L(s,~,r)
recipe (§1.4).
the following fundamental question:
§6.2
be an Euler product con-
How is the functional equation o f
Weyl symmetry.
L(s,~,r)
associated to the data
we construct certain distributions depending on
Theorem (6.2)
which
The problem we study here concerns
reflected in the structure of the 'explicit formula'
In
~A
~
and
r
{~,r}?
and in
we prove that these distributions possess a certain element of
These distributions had already been considered by Weil [ 42 ] for
a certain class of Euler products that had a definite number theoretic interest.
The idea of splitting the distribution obtained by Weil into two parts so as to
reflect both sides of the functional equation does not seem to have been considered
before~ our methods are nevertheless the same as those of Weil.
In a final section
we state a simple lemma that relates the location of the zeros (and poles) of
L(s,~,r)
to the positivity of the distributions that appear in the explicit for-
mulas.
Due to our incomplete knowledge about ramification phenomena present in
automorphic representations we are forced to introduce the following
Working Hypothesis:
~ = ~
fs an automorphic representation of the adele group
P
~A;
p
~
has a conductor set
f(~) = {p}
which consists of a finite set of primes
for which the local components are not class one representations;
complex representation of the dual group
character;
class in
for each prime
cG C
p
associated to
not in
~
f(~)
CG~
let
as in §1.4;
of dimension
{gp}
d
and
r
is a
X
is
its
be the semisimple conjugacy
for each prime
p
in the group
ci
G~
in
f(~)
we
P
assume the existence of a conjugacy class {g~}
which is the
199
dual group of a subgroup
r'
of
CG~
associate,
G'
of dimension
of
G
d' ~ d;
and of a finite dimensional
to the local component
~
representation
and
r
we
as in §1.4, a F-factor
d
r(s,~oo, r ) = ~ - 2 ( s - X i ) r ( ~ ) ;
i=l
we let
L(s,~,r)
= £(s,~ ,r) p e ~f(~) det(id,-p-Sr'(g~))-i p ~ f ( ~ ) det(Id - P - Sr(gp))-i
be the Euler product associated
L(s,~,r)
with the understanding
suitably modified.
have a meromorphic
functional
f(~,r)
that the local factors at primes
a number field
K
~n
f(~)
The key assumption about these Euler products
continuation
and
have to be
is that they
to the whole s plane and in fact satisfy the
= g(z,r) f (~,r)
s(~,r)
the functional
L(s,~,r)
only over
p
-s
L(l-s,z,r) ,
is a rational number all of whose prime divisors belong to the
f(~)
dK
(i d - p-Sr(gp))-i
equation
conductor set
where
{~,r}; for simplicity we write it as
= F(s,z , r ) ~ d e t
P
L(s,~,r)
where
to the data
is a complex number of absolute value
Assume
Over
equation probably has to be taken in the form
I~_ S
= £K(~,r)(d d i m r f(~,r)) "2 L(l-s,~,r),
is the absolute value of the discriminant
~.
I.
L(s,~,r)
of
K/~.
Here we will work
is the ratio of two entire functions
at most a finite number of poles and is bounded in every vertical
each having
strip of finite
200
width.
Finally we assume that the coefficients
~(s,~,r) = ~ d e t ( l
in the expansion
d-p-sr(gp))-I
P
co
=
satisfy for large positive
Z a(n)n -s
n=l
x
Z a(n) << x.
n<x
The only justification
Hypothesis
that we can offer for introducing
is that all known Euler products
algebraic geometry seem to satisfy it.
that arise in number theory and
In particular
§2.3
are of this kind and therefore
6.2.
The Explicit Formula and Weyl Symmetry.
We consider
real line
(A)
if
~
£
all the examples given in
the contents of Theorem
(6.2) are not empty.
of complex valued functions defined on the
that satisfy the following properties:
h(x) @ ~ ,
there is a real number
h(x)exp{(½+a')Ix I}
(B)
the class
the Working
h(x) C ~
is integrable
a' > 0
such that the function
on the real line;
and its derivative are continuous on the real line except at a
finite number of points
{~.}
where
h(x)
and
~(x)
have discontinuities
of the
1
first kind with
(C)
h(~i) = ½{h(~i+)
there exists a real number
h~x)exp{(½+b)Ix]}
= o<l)
The properties
as
+ h(~i-)};
b > 0
Ixl ÷
such that
and
~.
that characterize
use freely, as we will do in
h(x)exp{(½+b)Ix I} = o(i)
the class of functions
the following without
stopping
~
allows us to
to justify it, the
201
inversion formula for the Fourier transformation;
Lang's presentation
of distributions
of
Well's result
we will also use, following
([ 19 ], chap. X),
some of the properties
in the sense of Laurent Schwartz [ 34 ].
We now define the Mellin transform of a function
h E ~
by means of
the integral
R(s)
we will consider
h(s)
i
(s-½)tdt;
as a function of the complex variable
priately we could consider
algebra of rank
= /h(t)e
h(s)
s;
more appro-
as a function on the dual of a complex Lie
which is connected with the dual group
CG.
In order to
simplify our formulas we also consider the subclass of functions defined by
~o
= {he
: £(0)
= t{(1)
We now assume the Working Hypothesis
consider,
for positive real numbers
a
sists of the boundary of the rectangle
the line
Re(s) = ½
-T ~ Im(s) ~ T.
~0 = -a - iT
f
(a)
If
for the Euler product
T,
the contour
We let
a = l+a+iT,
~ = l+a-iT
be the corners of this rectangle;
sO = -a+
line joining
is not the imaginary part of a zero of the Euler product
Cauchy's residue theorem applied to the function
iT
and
and
~
we let
~
and
L(s,v,r),
h(S)d~ log L(s,~,r)
~.
then
gives
<T~(O)
IY _
L'
f h(s) ~ (s,~,r)ds -
(~)
located about
-a < Re(s) < i + a
and
and
which con-
for a complex number
taken along the straight
/ h(s) ~L' (s,~,r)ds = (2~i)
C(a,T)
L(s,z,r)
C(a,T)
in the s-plane symmetrically
and is defined by the inequalities
denote the line integral
T
and
: 0}.
f
L'
h(s) ~ (s,~,r)ds + o(i),
(~o)
202
where the sum
~
runs over the zeros (and poles)
p = B + iy
of
L(s,w,r),
Y
counted with their proper multiplicities and whose imaginary parts satisfy
IYl ~ T; the error term
o(i),
which results by integrating along the horizontal
rails and using the fact that the number of zeros
with
T < Imp < T + 1
tion
of
where
T
a'
is
<< log T,
which tends to
and
b
0
as
(and poles)
O
of
represents here and in the following a funcT ~ ~;
also we assume that
a < a' < b
are the real constants that appear in conditions
in the definition of ~ .
L(s,w,r)
From the Euler product definition of
(A)
L(s,W,r)
and
(C)
and the
functional equation we obtain
L'
d
T (s,w,r)ds = ~ s log L(s,w,r)
=
=
d log
ds
ddslog
~-rd
~
s-h i
I I ~-~(s-li)r(--2---)
d
+ ~ ~(s,#,r)
i=l
d
77i=l
~-½(i-s-~i)r(~-=) + ~d ~(s,~,~)
d
½--S
+ -71-_log f(~,r)
,
where
~l,...,%d are the infinity types that appear in the F-factor
F(l-s,~ ,~);
if we use the Fourier inversion formula we obtain easily
d
_h
f h(s) d log 77- ~ 2(s-%i) +
(i)
(~)
=
-
i=l
d
~
f ~(s) d log 77- ~-½(l-s-Xi)
(~0)
(dim r) log ~½ S ~(s)ds + (dim r) log ~
(~)
i=l
S
h(s)ds
(~0)
T
T
= (dim r) log ~½{ / h(½+it)i dt + S h(½+it)i dt + o(i)}
-T
-T
b
= 4~i (dim r) (log ~ ) h ( 0 )
From the Euler product
+ o(i).
203
~(x,~,r) = ~ d e t ( l d - p-Sr(gp))-i
P
we obtain easily, in the region of absolute convergence
~I
where
X
(s,~,r) = - ~ (log P)x(g~)P
n
P
is the character of the representation
positive powers of all the prime numbers.
r
-ns
,
and the sum runs over all the
If we observe that
s = 1 + a + it
and
ds = i dt, then we obtain
~,
^
/ h(s)~ (s,~,r)ds
(2)
(~)
f h(s)(- ~ (log p)X(g$)p-nS)ds
(~)
n
P
T
= - f ~(l+a+it)~ (log p)X(gp)e -(l+a+it)nl°gpidt;
-T
n
P
this last expression becomes, when we substitute the integral defining
h(s),
T
- f [ (log p)X(gp)e -(l+a+it)nl°g'p f h(x)e(½+a+it)Xdx idt
-T n
P
T
. . n . . . . (½+a+it)u-(l+a+it)nlog p du
= - f idt I f (log p)Xkgpjnku)e
-T
n
P
T
/ p -½n (log p)x(g$)h(u)e (½+a+it)(u-n log p) du;
= - -T
/ idt p~
if we make the change of variable
u + u + n log p,
then the last integral becomes
T
/
=_ -T
/ {[nmHP,n
P
(u) eitUdu}idt,
204
where
Hp,n(U) = p-½n(log p))<(pg)h(u + log pn)e(½+a)u.
Since
h(u) ~ £ 0
we have
iHp,n(U) I << p-½n(log p) ix(g~)le(½+a)Ue-(½+b)(u+nlogp)
<< (log p) Ix(g~)Ip-n(l+b)e-U(a-b);
similarly we obtain
~v
h(s)(- ~ (l-s,~,r))ds
/
O)
C~ 0 )
=-
T
_/T{~n ~ HP, n(u)eitudu} idt,
P
where
* (u) = p -~n (log p)X(g ~ )h(u - log pn) e (½+a) u
Hp,n
where
X
is the character of the contragredient representation
r; again using
the equality
T
- / dt /H(u)eiUtdu = - 2~H(0) + o(i),
-T
IR
we collect the results of formulas
i
2~i
(2) and (3)
in the formal identity
^
-½n (log p){x(g~)h(log pn) + ~(g~)h(-log pn)}
/ h(s)ds 6(s,~,r) = - [ p
C(a,T)
n
P
+ o(i).
205
The F-factors in
L(s,~,r), contribute, after a simple change of variable,
the formal identity
d
f
:r(:::i>
:og
C(a,T)
½+iT
d
2
d
-
½-iT
i=l
-
_ d log~r(l-s-%i)}
i=l
i=l
T
d
F' ( ½ ( ½ + i t - li) ) + ~ F' ( ½ ( ½ - i t - ~ i ) ) } d t .
= ~i / h(½+it) ~ {~
-T
i=l
To evaluate the last expression we are lead to consider the following
integral
: +iT
d
h(s) 77 :og F(as+b),
2~i
½-iT
where
a
is a positive real number and
change of variable
s = ½ + it
b
is any complex number.
it follows that this last integral is
T
i / ~(½+it)
2 i
-T
F'
a T (a(½ + it)+b)idt
= (2~
a
- ~' la,b ) + o(I),
where
h(t)
= h(½+it)
= / h(x)eltXdx,
IR
la,b(t) = ~ (½a + b + iat)
and
From the
206
(f,g)
Let us assume
for the moment
+Ia,b(t)
b
is real and put
r'
= ~ (½a + b + iat),_la,b(t)
+la,b(t)
then clearly
that
= If(x)g(x)dx.
= _Ia,b(t).
We also have,
r'
= ~ (½a + b - iat),
by a well known
formula
([
,
p. 13)
lim +la,b,M(t)
+Ia,b (t) = M÷oo
where
+Ia,b,M(t)
: log M + ~ ( t )
M
and
+gM(t)
= _
n=0
The Fourier
transformations
of
+~(t)
+gM(x)
= _
n+½a+b+iat
"
is
M
~
n=0
f
-ixt
e
n+½a+b+iat
dt
and hence
M
ixt
-k
+gM(_X ) = _ [ /
e
dt;
n=O 1R n+~"~a+b+iat
we now make the chan~e of variable
T = at, dT = adt
in the last integral,
obtain
ixt
ixT/a
/
e
IR/ e
]R nqS~aTb+ia t dt = - a
n~a+b+iT
dT ;
to
207
we use the classical distribution formula ([ 27 ], p. 430)
ixT/a
a
aT=
n+½a+b+iT
~f
a~
to finally obtain
27 -(n+~a+b)x/a
{y e
if
if
x> 0
x < O.
Observe that the last formula holds true without any assumption about the complex
nature of
b.
Finally
we o b t a i n
M
- ~ 2~ e-(n+½a+b)x/a
a
+gM(_X ) = { n=0
0
if
x > 0
if
x < O;
therefore we have
0
+gM (x) = { 2~ e(½a+b)x/a l-e (N+l)x/a
--•
a
l_eX/a
The Fourier transform of
+Ia,b,M(t)
if
x > 0
if
x
<
0.
is
+la,b,M(X) = ( 2 ~ 0) log M + +gM(x),
where
2~ 0
60
is Dirac's distribution concentrated at the origin, or equivalently
is the Fourier transform of the constant function
I.
transform of the function
Ia,b,M(t)
= log M + _gM(t),
where
M
_gM(t) = _ [
1
n=on+½a+b-iat
is
_la,b,M(X ) = ( 2 ~ 0 ) l o g M + _gM(x),
Similarly the Fourier
208
where
27 -(½a+b)x/a.l-e -(M+l)x/a
,
--- e
-x/a
_gM(x) = { a
l-e
0
From the Plancherel
if
x > 0
if
x < O.
formula
<_h ,_la,b ) = <h,_la,b >
we deduce
<_h ,+la,b> = <h, lim
M ÷ co -la,b,M>
= lira (h,_la,b, ~
M->oo
e_(½a+b)x/a.l_e-(M+l)x~
= lim {2~h(O) log M - f
M+ ~
0
2__~
a
h (x)d~
l_e-X/a
and
<_h
,
,+la,b >
=
f
-~
0 - - e(½a+b)x/a'l-e(M+l)x/a
2~a
i -ex/a
= lim {2~h(O)log M - f
M÷ ~
0
2~ e~½a+b)x/a.l-e(M+l)x/a
a
l_e-X/a
lim {2~h(O)log M
M * ~
_
h(x)dx}
•h(-x) dx}.
We have thus proved the following result:
Lemma 6.1.
number
a > 0
h
is a function in the class
and a complex number
Wa'b(h)
then we have
If
b
~
and if for a real
we define a distribution
Wa, b
= lim {/ e -(½a+b)x/a l-e-(M+l)x/a h(x)dx - ah(O)log M},
M~ w 0
1-e -x/a
by
209
lim { i
/ ½+iT fi(s)d
M ÷ ~ 2~i
½-iT
We now specialize Lemma (6.1)
i
½+iT^
2~i
log r(as+b) = - Wa, b (h).
to the case under consideration to obtain
d
d
-iT h(s>~s log i : l ~ r ( ~ )
= - i~iw½'-½%i
(h(t)) + o(i)
and
1
½+iT
- 2~---i/
d
h(s) d
l_s_~_i
log ~ F ( ~
½-iT
i=l
i=l ~' "2~i
Let us now go back to the automorphic
dimensional complex representation
correspond
a distribution
d
) = - ~ W~ _~.~ (h(-t)) + o(i).
and the finite
representation
z
r; to the local component
W(~ ,r)
whose value at a function
we make
h
in
£
is
given by
W(Voo,r)(h)
co
= lim {/ l-e-(M+l)2x
d
M÷oo 0
l-e -2x
"(i=l
[ e(%i-½)X)h(x)dx - ½ dh(0)log M}.
Observe that half of the contribution coming from the expression
incorporated
finite prime
W(~p,r)
into the definition of the distribution
p
W(~ ,r).
we make correspond to the local component
whose value at a function
W(~p,r)(h)
h
in
~
~
P
(i)
has been
Similarly for a
the distribution
is given by
co
-½n
n
= - ~ p
X(gp)(l°g p)h(log pn).
n=l
We add the local terms
W(~p,r)
corresponding
to all the primes,
including the infinite one, to obtain what in the following is called a
Zeta Distribution:
(4)
W(~,r) = ~W(~p,r).
210
The relevant notion that we must now introduce
form, which when applied to the distribution
W(v,r)
whose value at a function
h
W(~,r)
in the class
W(~,r)~(h(t))
gives a new distribution
£
is given by
= W(~,r)(h(-t)).
If we put together the contribution
f ~(s)d
(~)
is that of a Weyl Trans-
to the integral
log L(s,~,r)
which comes from the right hand boundary of the contour
C(a,T)
with the contri-
C(a,T),
and if we use the
bution to the integral
f h(s)d log L(l-s,~,r)
(aO)
which comes from the left hand boundary of the contour
obvious fact that the derivative
of
log g(~,r)
is zero and
1 ~U~/o)h(s)dlog f(~,r)½-s = h(O) log f(~,r) + 0(i),
2~i(
then, letting
T + ~,
Theorem 6.2.
~llin
transform;
dimensional
let
we have the following explicit formula:
Let
~
60
be a function in the class £ 0
be an automorphic
complex representation
Assume the Euler product
Let
h
L(s,~,r)
be the Dirac distribution,
the formula
(4)
and let
W(v,r) ~
representation
of the dual group
satisfies
let
CG~
of
and
and let
h
be its
G/A and
r
a finite
X
the Working Hypothesis
W(~,r)
be the distribution
be its Weyl transform.
of
§6.1.
defined by
We then have
h(o) = (W(~,r) + W(~,r)~)'h + (~0 log f(~,r))-h,
O
its character.
211
where the sum
E
runs over all the zeros (and poles) of
P
is the factor that appears in the functional equation
L(s,~,r)
and
f(~,r)
L(s,~,r) = e (~, r) f (~, r)½-SL(l-s,~,r) .
Remark 6.2.1.
where
y
The sum
Z
P
lim
~
T÷°°
I%,t<T
is the imaginary part of
Remark 6.2.2.
If
is to be understood as a symmetric limit
h(p),
p.
A(s) = ~-½sF(~)~(s)
is the set consisting of the elements
Z
p
~(~) =
1
and
is Riemann's Euler product and
~, then the explicit formula
Z W~'h,
~oE~
~(p):O
where
W = ~ Wp,
P
co
W "h = - ~ p-½n(log p)h(log pn),
P
n=l
~e-½X.l-e-(M+l)2x
Woo'h = lim {/
l_e_2X
.h(x)dx
M ÷~ 0
½h(0) log ~}
and
wl.(h(t)) = W'h(t), ~ ( h ( t ) )
= W'h(-t);
this explicit formula is to be contrasted with the constant term in the Fourier
212
expansion of the Eisenstein series
E(o,s)
for the group
SL20R) and
Foo = {(01 i): n E ZZ}
c(~,s)e (60(s)+p)H(O)
E(d,S) =
+
~*
mC~
A(s+l~2.Os(Iml)~.~½sV (2wlmle2OH(d~ 2~imn(O)+pH(~)
Iml
where
e pH(d) 0
1 ~(o))( °
e_PH(o))k(0)
o = (0
is the Iwasawa decomposition of
and~
~,
c(l,s) = i,
is the Weyl group of the Lie algebra
~
algebra of the subgroup of diagonal matrices in
A(s)
o(fl,s) = A ( s + l ) '
= ~
SL 2.
fi(s) = -s
which is dual to the Lie
Similar such interpretations
can be given for the symmetric term
W(~,r) W
for many Euler products
L(s,~,r)
where there is no ramification.
A deeper understanding of the Weyl symmetries present in the distributions
of the form
W(~,r) + W(~,r) ~
and their connection with the constant term of
Eisenstein series, will undoubtedly come from developing explicit formulas for
Euler products in several complex variables.
The following lemma, for whose demonstration we refer the reader to the
original article of Weil [ 41 ],
Lemma 6.3.
is of some interest.
Let the notation and assumptions be as in Theorem 6.2.
a necessary and sufficient condition for the Euler product
L(s,~,r)
Then
to be entire
213
and for all its zeros
O
to have real part
W(~,r) + W(~,r) ~ + 60 log f(~,r)
Re(p) = ½
is that
be a positive distribution in the sense of
Laurent Schwartz.
Remark 6.2.3.
The proof of this lemma was given by Weil for a type of
distribution defined in the space ~
symmetry;
which did not contain an element of Weyl
the necessary changes that must be made in his argument to apply to the
present situation are of a trivial nature and can be carried out easily.
Remark 6.2.4.
As is well known ([ 34 ], p. 131)
distributions of positive
type have many interesting properties of a hermitian character
Th~orSme XVII)
([ 34 ], p. 131,
which are comparable to the Weyl symmetry which is present in the
explicit formula of Theorem 6.2.; for these reasons we believe it is of some
interest to study in greater detail the structure of the distribution
W(~,r) + W(~,r) ~
and of similar distributions that can be constructed from the
constant term of Eisenstein series in which appear Weyl groups that contain more
than 2 elements;
it is also of some interest to look at these results from the
point of view of Bochner's theorem about distributions of positive type.
214
References
[1 ]
A. N. Andrianov, Euler products corresponding to Siegel modular forms of
degree 2, Uspehi Mat. Nauk 29(1974), 43-110.
[2 ]
A. Borel, Formes automorphes et s~ries de Dirichlet, Sem. Boarbaki,
No. 466, June (1975).
[3 ]
W. Casselman, On representations of
GL 2
and the arithmetic of modular
curves, Modular functions of one variable II, Springer L. N. 349 (1973),
107-142.
[4 ]
W. Casselman, On some results of Atkin and Lehner, Math. Ann. 201 (1973),
301-314.
[5 ]
J. Coates and A. Wiles, On the conjecture of Birch and Swinnerton-Dyer,
preprint (1976).
[6 ]
H. Davenport, Multiplicative number theory, Markham, Chicago 1967.
[7 ]
P. Deligne, La conjecture de Weil, I, Publ. I. H. E. S. 43(1974), 273-307.
[8 ]
K. Doi and H. Naganuma, On the functional equation of certain Dirichlet
series, Inv. Math. 9(1969), 1-14.
[9 ]
S. A. Gaal, Linear analysis and representation theory, Springer-Verlag,
New York/Berlin 1973.
[i0]
S. Gelbart, Automorphic forms on adele groups, Annals of Math. Studies 83
(1975), Princeton Univ. Press.
[ii]
S. Gelbart and H. Jacquet, A relation between automorphic forms on
GL(2)
and GL(3), Proc. Natl. Acad. Sci. USA, vol. 73 (1976), 3348-3350.
[12]
I.M.
Gelfand, M. I. Graev and I. I. Piateckii-Shapiro,
Representation
theory and automorphic forms, Saunders, Philadelphia 1969.
[13]
G. H. Hardy, Ramanujan, Chelsea, New York 1940.
[14]
H. Jacquet, Automorphic forms on
[15]
H. Jacquet and J. A° Shalika, A non-vanishing theorem for zeta functions
of
[16]
GL(2), part II, Springer L. N. 278, 1972.
GLn, Inv. Math. 38 (1976), 1-16.
H. Jacquet and R. P. Langlands, Automorphic forms on
GL(2), Springer L. N.
114, 1970.
[17]
T. Kubota, Elementary theory
of Eisenstein series, Kodansha Ltd., Tokyo 1973.
215
[18]
E. Landau, Uber die Anzahl der Gitterpunkte in gewissen Bereichen (Part II),
G~tt. Nachr.
(1915), 209-243.
[19]
S. Lang, Algebraic number theory, Addison-Wesley, Reading, Mass. 1964.
[20]
S. Lang, Elliptic functions, Addison-Wesley, Reading, Mass. 1973.
[21]
S. Lang, On the zeta function of number fields, Inv. Math. 12 (1971),
337-345.
[22]
R. P. Langlands, Base change for
GL(2):
The theory of Saito-Shintani with
applications, preprint, Institute for Advanced Study, Princeton, NJ, 1975.
[23]
R. P. Langlands, Euler products, Yale Univ. Press, 1967.
[24]
R. P. Langlands, Problems in the theory of automorphic forms, in Lectures in
Modern Analysis and Applications, Springer L. N. 170 (1970), 18-86.
[25]
W. Li, New forms and functional equations, Math. Ann. 212 (1975), 285-315.
[26]
H. Maass, Uber eine neue Art von nichtanalytischen automorphen Funktionen
und die Bestimmung Dirichletscher Reihen durch Punktionalgleichunge~, Math.
Ann. 121 (1944), 141-182.
[27]
W. Magnus, F. Oberhettinger and R. P. Soni, Formulas and theorems for the
special functions of mathematical physics, Springer-Verlag, New York 1966.
[28]
A. Orihara, On the Eisenstein series for the principal congruence subgroups,
Nagoya Math. J. vol. 34(1969), 129-142.
[29]
H. Rademacher, On the Phragm~n-Lindelof Theorem and some applications, Math.
Zeitschr. 72(1959), 192-204.
[30]
R. A. Rankin, Contributions to the theory of Ramanujan's function T(n)
and
similar arithmetic functions, I, II, Proc. Cambridge Phil. Soc. 35 (1939),
351-372; III, Proc. Cambridge Phil. Soc. 36(1940), 150-151.
[31]
R. A. Rankin, An a-result for the coefficients of cusp forms, Math. Ann. 203
(1973), 239-250.
[32]
I. Satake, Theory of spherical functions on reductive algebraic groups,
Publ. I. H. E. S. 18 (1963), 5-69.
[33]
I. Satake, Spherical functions and Ramanujan conjecture, in Proc. Symp. Pure
Math., 9 (1966), 258-264, A. M. S. Providence R.I.
[34]
L. Schwartz, Theorie des distributions, vol. II, Hermann, Paris 1959.
216
[35]
A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric
riemannian spaces with applications to Dirichlet series, Jo Indian Math.
Soc. 20 (1956), 47-81.
[36]
F. Shahidi, Functional equation satisfied by certain L-functions, preprint
(1976).
[37]
Go Shimura, Introduction to the arithmetic theory of automorphic functions,
Iwanami Shoten and Princeton Univ. Press 1971.
[38]
G. Shimura, On the holomorphy of certain Dirichlet series, Proc. London Math.
Soc. 31 (1975), 79-98.
[39]
A. M. Stark, Some effective cases of the Brauer-Siegel Theorem, Inv. Math.
23 (1974), 135-152.
[40]
A. Weil, Ueber die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, Math. Ann. 168 (1967), 149-167.
[41]
A. Weil, Sur les "formules explicites" de la thEorie des nombres premiers,
Comm. S~m. Math. Universit~ de Lurid (dedi~ ~ M. Riesz)
[42]
(1952), 252-265.
A. Weil, Sur les formules explicites de la theorie des nombres, Isv. Acad.
Nauk. 36 (1972), 3-18.
Oscillations of bases in n u m b e r
theory and c o m b i n a t o r i c s
M e l v y n B. N a t h a n s o n
D e p a r t m e n t of M a t h e m a t i c s
S o u t h e r n Illinois University
Carbondale, Illinois 6Z901
I.
Let
]N denote the nonnegative integers, a n d let B ~ ] N
the set of all s u m s
then
B
of h
B
order
if h B = IN,
h
order
hB
h . If h B
minimality,
order
n
B
if B
but for every
e l e m e n t s of
is a m i n i m a l
B' ~
B.
b e B
gave e x a m p l e s
of a s y m p t o t i c bases of o r d e r
a s y m p t o t i c basis of o r d e r
Z.
exist a s y m p t o t i c bases
of o r d e r
ISI < 00, then
B\S
B
Similarly,
h ; that is,
there are infinitely
h
contains a m i n i m a l
basis of
that there exist
a s y m p t o t i c bases, a n d N a t h a n s o n [i0] constructed e x a m p l e s
for all h_> Z.
basis of
StBhr [IZ] introduced this idea of
and p r o v e d that every basis of o r d e r
h
denote
If h B = IN ,
is a n a s y m p t o t i c basis of o r d e r
such that n ~ h(B\~b}).
a s y m p t o t i c b a s e s of o r d e r
hB
is an a s y m p t o t i c basis of
h . H~[rtter [8] s h o w e d by a nonconstructive a r g u m e n t
minimal
B\S
B
h
contains all sufficiently large n u m b e r s ,
integers
h . T h e set
but hB' / IN for every p r o p e r subset
a s y m p t o t i c basis of o r d e r
h , but no p r o p e r subset of
many
Let
contains all but finitely m a n y
is an a s y m p t o t i c basis of o r d e r
is a m i n i m a l
.
not necessarily distinct e l e m e n t s of B .
is a basis of o r d e r
I'4, then
B
Introduction
of m i n i m a l
H~rtter [8] and N a t h a n s o n [I0] also
Z that do not contain any m i n i m a l
Indeed, E r d B s and N a t h a n s o n [3] s h o w e d that there
Z
such that, for e v e r y subset
r e m a i n s an a s y m p t o t i c basis of o r d e r
is no longer a n a s y m p t o t i c basis of o r d e r
basis that does not contain a m i n i m a l
Z.
Z,
Clearly,
S~
but if
B
B , if
ISI = o0, then
is an a s y m p t o t i c
a s y m p t o t i c basis.
T h e r e is no classification of m i n i m a l
a s y m p t o t i c bases, nor is there any
s i m p l e criterion to insure that a n a s y m p t o t i c basis contains a minirr~l a s y m p t o t i c
basis.
minimal
of o r d e r
E r d • s a n d N a t h a n s o n [6] p r o v e d that the square-free n u m b e r s
a s y m p t o t i c basis of o r d e r
Z of s q u a r e - f r e e n u m b e r s
contain a
Z and also that there is an a s y m p t o t i c basis
no subset of w h i c h is m i n i m a l .
But it is usually
difficult to d e t e r m i n e w h e t h e r or not a given a s y m p t o t i c basis contains a m i n i m a l
a s y m p t o t i c basis.
For example,
the set B
by L a g range's t h e o r e m , but it is not k n o w n
basis of o r d e r
Z.
= {mZ+n Z}~,n=0
if B
is a basis of o r d e r
contains a m i n i m a l
asymptotic
Z
218
T h e r e also exist asymptotic bases
the sense that, if S ~ B ,
bu£ an asymptotic
1-minimal
order
nonbasis
bases
Let
are
B
If
of order
B
of order
Z
but
= ~
for
every
if
ISI > r
the minimal
set
basis o f o r d e r
(Erd~s
of order
basis
superset
h,
and
then
of order
]3 is a maximal
proper
Z that are r - m i n i m a l in
Z if ISI < r
Nathanson
[3]).
The
bases.
is not a basis
h . The
of order
is a n a s y m p t o t i e
is not an asymptotic
nonbasis
hB'
B\S
precisely
B~I>4.
h . If
then
B
h,
B.
is a nonbasis
then
nonbasis
B' ~
B
B
is an asymptotic
of order
Similarly,
of
h
B
if hB/
~,
is a maximal
t
asymptotic
for
B
nonbasis
every
proper
superset
is a maximal
infinite
of order
h
B' ~
asymptotic
sequence
if
n ~hB
B
and
nonbasis
of numbers
for infinitely
all sufficiently
of order
not belonging
Z,
to
n,
large
and
ZB,
many
n.
but
n ¢ hB'
In particular,
if
if n I < n 2 < n 3 < ...
then
n. - a ~ B
is the
for every
i
nonnegative
integer
a ~ B
and
all sufficiently
large
n..
Nathanson
[i0] introduced
i
this idea
of maximality,
nonbasis
of order
h
for every
h > Z
which
was
the union
asked
several
ErdBs
proved
every
a class
thatif
B~]N
and
Z.
This
tic nonbasis
implies
also
in the sense
that,
order
ISI <
Z
IS] >_ r
if
nonba
of density
is an asymptotic
is contained
that if
B
has
d~j(ZB}
density
< l,
then
each
of
Finally,
been
and
of arithmetic
B
he
answered
Turj~nyi.
examples
of maximal
progressions,
and
nonbasis
of order
also
Z
for
in a maximal
asymptotic
d(B)
if the surnset
B
= 0
and
is a subset
nonbasis
ZB
of a maximal
asympto-
Z.
exist asymptotic
if S~]N\B,
r,
but
nonbases
then
B~
~131 proved
zero,
of order
is best
subsequently
I-le constructed
h,
progressions.
"nontrivial"
then
h.
of order
Nathanson,
that every
S
B
BUS
becomes
of order
Z
that are
is still an asymptotic
an asymptotic
basis
r-maximal
nonbasis
of order
of
Z
if
T h e l - m a x i m a l nonbases are precisely the
s es.
Turj~nyi
result
F
(Erd}Js and Nathanson [3]).
maximal
nonbases
B~
density
of order
There
not unions
if
F ~ ,
asymptotic
that have
Hennefeld,
showed
of order
nonbases
of aritl~netic
[Z] constructed
that were
and
nonbasis
asymptotic
nonbases
of ErdBs,
Nathanson
finite subset
upper
about
nonbases,
of a maximal
of maximal
results
and
the maximal
of a finite number
nonbases
of order
has
is a subset
questions
by the following
asymptotic
classified
and
Nathanson
Z
possible.
whose
that there
exist maximal
[ii] constructed
counting
functions
asymptotic
a class
have
of "thin"
order
nonbases
maximal
of magnitude
of order
asymptotic
~-.
This
2
219
PIennefeld [9] constructed the first e x a m p l e of an asymptotic nonbasis of
order
Z that cannot be e m b e d d e d in a m a x i m a l
asymptotic nonbasis of order
Z.
E r d ~ s and N a t h a n s o n [5] constructed a class of "thin" asymptotic nonbases of
order
Z such that each set in this class had
O(~x--)
elements not exceeding
and such that no set in this class is contained in a m a x i m a l
order
Z.
asymptotic nonbasis of
E r d B s and N a t h a n s o n [6] proved that there does not exist a m a x i m a l
asymptotic nonbasis of order
Z consisting only of square-free n u m b e r s ,
that there does exist an asymptotic nonbasis
such that B ~ J {q}
number
x,
B
of order
is an asymptotic basis of order
but
Z of square-free n u m b e r s
Z for every square-free
q ~ B.
Finally, it is possible to partition ~
that A
into two disjoint sets A
is a m i n i m a l asymptotic basis of order
nonbasis of order
"random"
Z; m o r e o v e r ,
elements are m o v e d
Z and
B
and
is a m a x i m a l
B
such
asymptotic
this partition can be constructed so that, as
from
A
to B
to A
to B ...,
f r o m basis to nonbasis to basis to nonbasis . . . and the set
eously f r o m nonbasis to basis to nonbasis to basis...
B
the set A
oscillates
oscillates simultan-
(Erd~s and N a t h a n s o n [4]).
It is not k n o w n w h i c h of the results above are true for bases and nonbases
of orders
h > 3.
In this paper I consider a combinatorial analog of m i n i m a l bases and m a x i m a l
nonbases.
Let
,~(~)
denote the collection of all finite subsets of ~ ,
~ ,~(~) . Denote by
h~
distinct sets belonging to
order
h . Otherwise,
but finitely m a n y
order
then
h.
~
~
the collection of all unions of h
~.
If h ~
=~(]N),
~
is a union nonbasis of order
elements of <~(IN),
then
But if there are infinitely m a n y
sets in
I~rdds [I] obtained results ab()ut union bases for
bases and m a x i m a l
h . If h ~
contains a]l
<~(I~)
~(]N)
not belonging to h ~ ,
h . Recently, D e z a and
analogous to k n o w n results
I shall consider m i n i m a l asymptotic union
asymptotic union nonbases for the collection of finite subsets
of the natural n u m b e r s .
It is easy to prove that every union basis of order
contains a m i n i m a l union basis of order
of order
is called a union basis of
~3 is an asymptotic union basis of
is called an asymptotic union nonbasis of order
about bases in additive n u m b e r theory.
~(~q)
then
and let
not necessarily
h for ~7~(ix[) is contained in a m a x i m a l
h,
h
for
and that every union nonbasis
union nonbasis of order
h.
But
p r o b l e m s about asymptotic union bases and asymptotic union nonbases are m o r e
complicated.
For
h > Z there is a trivial construction of m i n i m a l asymptotic
union bases of order
h.
asymptotic union bases.
For
h = Z I shall construct a class of nontrivial m i n i m a l
But it is not true that every asymptotic union basis of
220
order
h
contains a m i n i m a l asymptotic union basis of order
an asymptotic union basis of order
for every
h.
I shall construct
Z, no subset of w h i c h is minimal.
h > Z there exist asymptotic union nonbases of order
e m b e d d e d in m a x i m a l
asymptotic union nonbases of order
k n o w n if there exists a m a x i m a l
h.
h
Similarly,
that cannot be
Indeed, it is not
asymptotic union nonbasis of order
h for any
h>2.
Notation.
T h e natural n u m b e r s
case letters denote natural n u m b e r s
natural n u m b e r s .
iN are the nonnegative integers.
and capital R o m a n
letters denote sets of
Capital script letters denote sets of sets of natural n u m b e r s .
interval of integers
a < n<
b is denoted
the set of all finite subsets of S.
[a,b].
If S ~ ] N ,
then
B. ¢ ~ .
1
Then
h~
for i = i, Z ..... h.
(~.
h~
denote ~
~J ~ J
then the
B I U B Z ~ .. ~ ~h'
... ~ J ~
(h times).
1
consists of all unions of h
Clearly, ~5 Z Z S ~
in
Let
The
c~(S)denotes
If ~ i ~- ~;~(]~q) for i = i, Z ..... h,
union set ~i<3 ~Z<; ... ~ ~h consists of all sets of the form
where
Lower
3@C_ ....
Finally, let
IX[
not necessarily distinct sets belonging to ~ .
Let e \ ~ 3
denote the relative complement of
denote the ca rdinality of X.
221
Z.
Minimal
A union basis
~
union nonbasis of order
of order
union bases
of order
and maximal
h
,~(IN)
for
h for every
Bc ~.
h is m a x i m a l if ~ [.J {A}
union nonbases
is m i n i m a l if ~ \ { B }
is a
Similarly, a union nonbasis
is a union basis of order
h for every
A ~..,~(l~)%k~ . In this section I prove that every union basis contains a m i n i m a l
union basis and every union nonbasis is contained in a m a x i m a l union nonbasis.
THEOREM
union
basis
I.
Every
of order
Proof.
union
basis
of order
h
for
~(~)
contains
a minimal
h.
Let
= {
~
B i}i=l
0o
be a union basis of order
h for ,~(IW).
If
is minimal, w e are done.
Otherwise
B.} is a union basis for s o m e B. ~ ~ .
1
1
Let iI be the least subscript such that ~ i =(~\{ Bi I} is a union basis• If ~ l
is minimal, w e are done.
Otherwise, let iZ be the least subscript such that
~2
= ~i ~{ Bi Z} = ~ \ { Bi I' Biz}
~{
is a union basis.
Continue this process inductively.
If it stops after a finite n u m b e r of iterations, then the last union basis in the sequence
is minimal.
~
Otherwise, there is an infinite decreasing sequence of bases
~i~2__~
....
where
~k
: ~k-l~{Bik }
and il < iz< i3< ....
is the least integer greater than ik_ 1 such that ~ k _ i k { Bik}
I claim that ~':'~:~k°°:l ~ k
:~\{
Bik}k:l
Let X ~ ,~ (IN). T h e set X
i<i _ r if B i ~ _ X
Bt Z'" • . ' Bth~
Btj (
~r'
and
B.. ~i~
Since
~r
h~*.
Thus,
If ~ * \ { B i} is a basis for s o m e
and ~k_iX{Bi}
Therefore,
is a union basis•
~ *\{B.}
THEOREM
Z.
Proof.
h~.
Let
Let ~
~'"
tj ~i k for all k <_ r since
Btj~X
. Therefore,
is a union basis oforder
B.i ~ ~ * '
is a nonbasis for e v e r y
Btj ~ ~ *
h.
then ik_ 1 < i < ik for s o m e
]B. ~ ~:"~,
a n d so ~3;:" is a m i n i m a l
i
h for ,Y~(JN).
Every union nonbasis of order
in a m a x i m a l union nonbasis of order
X/
i such that
r
is abasis, thereare sets Btl,
But this contradicts the minimality of ik.
1
union basis of order
h.
is finite, so there is an integer
and also t.] / i k for all k > r since
and so X ~
ik
is a union basis.
is a minimal union basis of order
~ r such that X = ~ hj:l Bt'"
J Clearly,
for j :i ..... h,
and
is contained
h.
be a union nonbasis of order
~i = ~ ~
h for .~(IN)
(~(]N)\,~(X)).
Clearly,
h.
Choose
X ~ .~(IN) with
~ 1 is a nonbasis since
k,
222
X~
h ~ I.
so
~
l
But ~ I
contains all but a finite n u m b e r of elements of ~(~q),
is contained in a m a x i m a l union nonbasis of order
h.
and
223
3.
M i n i m a l asymptotic union bases
A n asymptotic union basis
there are infinitely m a n y
(~
of order
h
is m i n i m a l if for every
sets in ..~ (Eq) that do not belong to h ( ~ \ { S} ). A n
asymptotic union basis of order
1 is simply a co-finite subset of .~(Eq).
no m i n i m a l asymptotic union basis of order
1 exists.
there do exist m i n i m a l asymptotic union bases of order
that every asymptotic union basis of order
union basis of order
h.
h
3.
h > Z,
h.
Let h >
Z,
Clearly,
however,
But it is not true
h > Z contains a m i n i m a l asymptotic
~
such that
Z for every finite subset ~
is an asymptotic union nonbasis of order
THEOREM
For
Indeed, I shall construct a basis
is an asymptotic union basis of order
~\~
S ~
~,
~\
2
but
Z for every infinite subset
and let TI, T z , . . . , T h
be a partition of IN into
n o n e m p t y sets at least two of which are infinite. T h e n ~_jh
i=l
h.
£/(Ti)\{ ~ } )
is
a m i n i m a l asymptotic union basis of order
Proof.
Let ~
for j = l,...,k,
=~_Jhi=l ~ ( T i ) \ { ¢ } )"
then
X(~Tij ~ ~
Let X ~ - ~ ( ] N ) , X / ¢ .
for j =i ..... k,
If X ( - ~ T i . / ¢
J
and
k
X = k_j ( X ~ T i . ) ~ k ~
j=l
hence h ~
=,~(Eq)~{~}
•
~h~
3
M o r e o v e r , if X ( ~ T . / ~ f o r each i,
i
then X =~jh
i=l
(Xf'-~Ti) is the unique r e p r e s e n t a t i o n of X as the union of h e l e m e n t s of ~
Let
S ~ g
,
say,
S ~ ~(TI)\{~
} . At least two of the sets
.
T.l are infinite,
hence ~jh
T. is infinite, and so there are infinitely m a n y sets X ~ "~r(]N) such
i=Z i
that X~-%TI = S and X~-~T.~ l/
for all i = Z, 3 ..... h. But X / h ( S \ { S } ) ,
and
so
~kk{S}
is an asymptotic union nonbasis of order
E a c h m i n i m a l asymptotic union basis
~
h.
This proves the T h e o r e m .
constructed above has the
property that if B ~ (~ , then every n o n e m p t y subset of B
the "trivial" m i n i m a l asymptotic union bases.
is in ~
T h e following L e m m a
.
T h e s e are
will be
applied to construct a class of nontrivial m i n i m a l asymptotic union bases of order
Z,
and also to construct union bases of order
asymptotic union bases of order
LEMMA.
Z that do not contain any m i n i m a l
Z.
Let Rk be a n o n e m p t y s u b s e t of
T h e n there is a family
~k+l
[1, nk] ,
and let n~+ l >__nk + 3.
of subsets of [i, nl~+l] with the following properties:
224
(i)
[nk + I, nk+l] ( /$k+l'
and
B(-~[nk + I, ink+l] / 6
(ii) If X C [I, nk+1] and Xf-h~[nk + I, n]<+l] / ~,
(iii) If Ph<U Ink + l, 5<+1 ] : B I U B z, where
then either B I = 1%k and
B Z =[n k + i, nk+l]
(iv) If IR.kl > I, then ~k+1
or
for every
B c (~k+l;
then X ~ Z(~k+1 u
{P~k});
BI, B z ~ ~9k+l<J.~([l, nk]),
B Z = P~k and
B 1 = [nk + I, nk+l];
does not contain every nonempty subset of
[nk + 1, nk+l].
Proof.
Case~: Suppose
i : [ % + 1, % + j
I suchthat
}Jl = m}.)
the f o r m
where
Since Izl : % + i - %->
Z~ =.)~(I)\{~,I}.
If lit = Zm,
fix x ~
3, there is
s'V
I, where
S' ~ S ,
1%' ~ 1%, S' C S , and
hence
nk+l]
2
a family
of subsets of
: {JC_I I I__< IJ I _<m}.
= { J ~ I I i__< IJl < m -
l } V {J_~IIx{ J and
and, second, those ofthe form
J E ~.
~k+l
Clearly,
for any J ~ #
JI, Jz ~ ~
. Since
and 1%Z~ 1%. Then
s'U
J,
U {I}
satisfies conditions (i) and (iv).
with XI = X(-~I / ~ . Let i%' = X(~it and S' = X(-]S.
then X : R U (S' U I) ~ Z ( ~ k + i V
{%})
R'U
I( (~k+ 1 and X ( - ~ I ~
X = Itl V S I <_) X I. Suppose that X I : I. Then
Z(~k+iU
and
Let (~k+l consist of the following two types of sets: first, those of
Let X ~ [ l ,
some
Let R : ~ , S : [ I , % ] \ R ,
(If Ill = 2 m + i, let ~
I andlet ~
for all X ~ ~k+l'
Then
I~I>I"
{Rk});
if R' ~
It,
Jl ~ ~ k + l
X : (1%1U S' U I i ) U
and 1%Z u
If I{l = It'
then X : (It' <J J ) U (S' U I)
. Suppose that X I / I. Then
IR_kl > I, we canwrite
1%1v S' Y
S' <J I ( (~k+l"
X I : Jl u Jz for
It' : 1%1[.J 1%Z' where
Itl~ i%
JZ C (~k+l' hence
( i t z U JZ ) : 1%' U S' ~ X I
This proves (ii).
Finally, let 1%U I = B I U
B Z for s o m e
B.(~I/If
for i = 1, Z, then Bi(-~I~ ~
for s o m e
I, say,
and
B1 = L
{ ~ } . But I ~ Z7 .
i : i, and so B l : S' Y L
It follows that i~ c B Z.
B Z ( ,~([i, nk]), and
n.k]). If
Therefore,
B.~-~I
I i =
But (itU l)(-~S : ~ , and so S l :
But this is impossible if B Z ¢ ~k+l'
hence
B e : 1%. This proves (iii).
Case II: Suppose
sets of the f o r m
V
BI, B Z ~ ~ k + i U J ( [ l ,
I%1
: 1, say,
S' Y I , where
%
: { r }.
Let ~k+l'
S' _~S = [i, nk]X{r}
consist of all
and I' _~I, I' / ~.
225
Clearly,
~k+l satisfies (i).
Let X C [1, nk+l] with X(~I # ~ . If r ~ X, then
X ( 6 k + i C Z ( ~ k + l ~ J {P~k} ). If r e X, then X~{r} ~ O~k+ 1 and X : (X~{r})U
{ r]. ¢ Z(~k+IU {R.k}). Thus, ~ k + I satisfies (ii). Finally, if P~kU I : B1U BZ,
where
BI, Bz ( O~k+iU~([l, nk]), then r e B.I for some
BI ~ ~k+l'
hence
BZ e ~k+l
and
B I ~ [i, nk] and so B I = { r} = R-k.
B Z = I. Thus, ~ k + l
i, say, i = I. Then
Then
I C BZ,
hence
satisfies (iii). This completes the proof of
the L e m m a .
THEOREM
4. There exist nontrivial minimal asymptotic union bases of
order Z.
THEOREM
that, if # C
5. There exist asymptotic union bases
8,
then
~\~
~
of order Z such
is an asymptotic union basis of order Z whenever
I ~ I < o0, but an asymptotic unionnonbasis of order Z whenever
particular,
~
In
does not contain a minimal asymptotic union basis of order Z.
Proofs. Let {nk}
for all k > l .
I~[ =~.
be a sequence of positive integers such that nk+ I > ~k + 3
I first construct inductively a sequence of sets ~ k ~ ( [ l ,
nk])\{ ~ }
and sets P~k ~ U k
~i" Let ~i =L~([I, nl])\{ ~ ]
Suppose that ~l'"
~ k and
i=l
.
.
.
.
i~i,. . . , Rk_ 1 have been determined. Choose any Rk ~ uki=l ~i" Let ~k+l ~-([I, nk+l])\~([l, nk] ) satisfy conditions (i)-(iv)of the L e m m a .
o0
UM=I @k"
Clearly,
~ ~ Z~
then X ¢ ~ i : Z~l--~Z(~"
some unique k > l .
Z ~ =~(~)'N{~ },
since ~ { ~ . Let X ~ ~-~(]m), X / ~ . If x C
Otherwise,
~
X E Z(~kll~{Rk})~- Z~.
for
Thus,
is an asymptotic union basis of order Z.
Let B_kU Ink + 1, nk+l] : BlkJ B2, where
B1, B Z ~ 03 . Since R k (
uki=l ~3iC~([l'-- nk]), it follows that BI, BzC[I,_ %+i].
implies that BI, BZ ( <jk+li=l~k+l C_ ~k+l U ~([l, nk]).
Zemma,
[i, nl] ,
XC[I, nk+l] and X~[nk+l, nk+l] / 6
By condition (ii) of the g e m m a ,
and so
Let ~ :
either B 1 : P~k or B Z : P~k" Thus,
Condition (i) of the L e m m a
By condition (iii) of the
P~kU [nk + I, nk+l] ~ Z(~\{P~k}).
The sets P~k~J [nk + i, nk+l] are pairwise distinct, although the sets R k themselves
need not be distinct.
H o w shall we choose the sets P~k ?
infinitely often as an Rk; that is, if B ~ ~
Then
RkU
Suppose that every set B ~ ~
, then B = P~k for infinitely m a n y
[nk + l, nk+l] ~ g(~\{ 13} ) for infinitely m a n y
minimal asymptotic union basis of order Z. Since
IB I > i, condition (iv) of the L e m m a
T h e o r e m 4.
is chosen
implies that
k,
k.
and so 03 is a
03 contains sets B with
~3 is nontrivial. This proves
226
Now
~C
suppose that every set B ~ ~
~ " If R.k ~ ~,
asymptotic
union
then
nonbasis
w e have [_JOOk=t~ k C- ~ \ ~ 2
is chosen exactly once as an Rk.
P,_k~J [n k + i, nk+l] j Z(~),
o~ order
and
Z whenever
P,-k ( 0 ~ \ ~
and so ~\~
r"fl =~"
for all k >t._
X(-h[n k + I, nk+[] / ~ . By condition (ii) of the L e m m a ,
if k > t ,
~\~
and so
Z(~\~ 0) contains all but finitely m a n y
is an asymptotic union basis of order
Theorem
5.
But if
l~r
Let
is an
<~,
Let XC[I,_ nk+1]
then
with
X ~ Z(~k+l<-) {R-k])~- Z(~{~)
elements of ,2~(IN). Thus,
g whenever
[~I <°° . This proves
227
4.
M a x i m a l asymptotic union nonbases
A n asymptotic union nonbasis
an asymptotic union basis of order
union nonbases of order
(~
of order
h for every
S ec~(Eq)\~.
is
T h e asymptotic
1 are precisely those 03 C-,~(]N) such that o~(]N)\~ is
infinite. T h e r e is clearly no such m a x i m a l
union nonbasis of order
nonbases of orders
h is m a x i m a l if B~J {S}
~,
hence no m a x i m a l asymptotic
i. It is not k n o w n if there exist m a x i m a l asymptotic union
h > Z.
But it is possible to construct asymptotic union nonbases
that cannot be e m b e d d e d in m a x i m a l asymptotic union nonbases.
THEOIREM
6.
then for every k > l
Proof.
S ~ h~
If
If
there are only finitely m a n y
~
sets X ~ h ~
sets X ~ ,2~(~) with
also contain all but finitely m a n y
COROLLARY.
Let
~
sets X ¢ j~(]N) with
m a x i m a l asymptotic union nonbasis of order
where
~
is
Sup posethat X ~
implies that h ~
X ¢ h~ Ch~
nonbasis of order
basis of order
S, T { ( ~ ,
h~
must
.~
h
such
cannot be e m b e d d e d in a
h.
~
IXl >t. H S _ C ~ ,
with
sets X
h,
with
then T h e o r e m
6
Ixl < t. ]But
is an asymptotic union basis of order
h.
proves the Corollary.
~t = {Xc~(Eq)
Ilxl > t}
is an asymptotic union
h that is not contained in any m a x i m a l asymptotic union nonbasis
7.
Let h _> Z, and let ~
be a m a x i m a l asymptotic union
h that is also an asymptotic union nonbasis of order
Zh - Z.
non-
If
then S(-~T : ~ .
Proof.
By m a = ~ m a l i ~ , both
bases of order h.
Xc
But the
h.
THEOREM
and
Therefore,
Ixl = k.
forall X ~ ( ~ I
if IXl > t. Therefore,
In particular, the set
of order
ha
Then
contains all but finitely m a n y
This contradiction
IX[ = k.
Therefore,
m a x i m a l asymptotic union nonbasis of order
a
h and
be an asymptotic union nonbasis of order
contains all sufficiently large sets.
Proof.
is S.
h,
IxI = k.
and so 0"5 (,9 {S } is an asymptotic union basis.
contains all but finitely m a n y
only k- element setin h(0~[.J { S } ) t h a t is not in h03
that h @
with
is a m a x i m a l asymptotic union nonbasis of order
, then S ~ 03
h(0"~J {S})
~3 is a m a x i m a l asymptotic union nonbasis of order
~ U { S} and 45 U { T}
Therefore, there i s a s e t
h((~L._){T}),
but X {
(Zh-Z)(~.
X /S,T
Then
are asymptotic unio~
such that X~ h ( ~ U {S})
X{hO~,
and so
x : sU BzU ... U Bh : TU B~U ... U B~,
228
where
B., B! ¢ ~
i
and so
for
X = BzU
. . ~J. B h.U . B.~ U
COI~OLLAI~Y.
S,T¢ ~ ,
i = Z ..... h.
If S~-~T = ~ , then
TC
BzU
... V
Bh,
1
then
Let
~3
s~T/¢.
U
B hI ~ (Zh-Z)~
be a m a x i m a l
But this is a contradiction.
asymptotic union nonbasis of order
Z.
229
5.
i.
Let
~
~(~)\~.
Open problems
be an asymptotic union nonbasis of order
Then
~
is r - m a x i m a l if ~ L_J ~
h,
and let
is an asymptotic union nonbasis
of order
h whenever
I~I < r,
but ~ ~ _ J ~
of order
h whenever
i#I > r.
T h e 1 - m a x i m a l asymptotic union nonbases are
precisely the m a x i m a l
asymptotic union nonbases.
asymptotic union nonbases of order
case
r = 1 and
2.
Then
~
Let
b e c o m e s an asymptotic union basis
h?
D o there exist r - m a x i m a l
This is not k n o w n even in the simplest
h = 2.
(~
be an asymptotic union basis of order
is r m i n i m a l
h,
and let ~ _ ~
if ~3\~f is an asymptotic union basis of order
.
h whenever
I.#l < r,
but ~ \ 2
I~I > r.
T h e l-minimal asymptotic union bases are precisely the m i n i m a l asympto-
tic union bases.
all r > Z and
order
b e c o m e s an asymptotic union nonbasis of order
~
h whenever
D o there exist r - m i n i m a l asymptotic union bases of order
h > 2?
h for
A r e there nontrivial m i n i m a l asymptotic union bases of
h > 3?
3.
Classify the m i n i m a l asymptotic union bases and m a x i m a l
union nonbases.
asymptotic
A r e there general criteria that imply that an asymptotic union
basis contains a m i n i m a l asymptotic union basis or that an asymptotic union nonbasis is contained in a m a x i m a l
4.
that ~
asymptotic union nonbasis ?
Is there a partition of .~(]N)
into two disjoint sets
is a m i n i m a l asymptotic union basis of order
asymptotic union nonbasis of order
5.
Z and
~
03
and
6~
such
is a m a x i m a l
2?
If w e consider intersections of sets instead of unions of sets, then w e
find a n e w series of u n a n s w e r e d combinatorial p r o b l e m s about
define an asymptotic intersection basis of order
such that all but finitely m a n y
sets in
~(l~)
of h not necessarily distinct sets in ~
bases exist?
h
.~(]N). F o r example,
for ,.~'(]N) to be a set ~
~(]N)
can be represented as the intersection
. D o m i n i m a l asymptotic intersection
D o e s every asymptotic intersection basis for
~(l'q)
contain a
m i n i m a l asymptotic intersection basis ?
6.
Then
B
Let
Q
be the set of square-free positive integers, and let B ~ Q .
is an asymptotic
LCM
basis of order
h
for Q
if all but finitely m a n y
square-free integers can be represented as the least c o m m o n
of B.
Similarly,
B
is an asymptotic
GCD
basis of order
multiple of h
h for Q
elements
if all
sufficiently large square-free integers can be represent.ed as the greatest c o m m o n
230
divisor
of h
elements
of
B.
We define
LCM
and
GCD
bases,
nonbases,
and asymptotic nonbases similarly.
Combinatorial t h e o r e m s about union and
intersection bases and nonbases for
._~(~xI) are equivalent to multiplicative t h e o r e m s
about
LCM
and
GCD
Z = P0 < Pl < PZ < "'"
q : c~(]N) -~ Q
LCM
by
bases and nonbases for Q
be the sequence of p r i m e s in ascending order.
q(B) = IIb~B qb
[q(B l)..... q(Bh) ] and
follows that ~ ~.~(IN)
nonbasis) of order
asymptotic
LCM
h
in the following way.
for all B ¢ ._~(]N). T h e n
q(Bl(-~... (-~B h) -- G C D
Let
Define
q(BiKJ ... [-J B h) =
(q(B I)..... q(Bh) ). It
is an asymptotic union (resp. intersection) basis (resp.
for
.~(]N) if and only if q((~) : {q(B)[B ¢ (~} _ C Q
(resp.
GCD
) basis (resp. nonbasis)of order
Thus, combinatorial t h e o r e m s for
.~(]N)
is an
h for Q.
can be translated into multiplicative
t h e o r e m s for Q.
It is natural to consider
set of al_~lpositive integers.
asymptotic
LCM
(resp.
elements of B.
W e define
nonbases similarly.
plicative n u m b e r
T h e set
GCD
integer is the least c o m m o n
LCM
and
B
GCD
bases and nonbases for the
of positive integers will be called an
) basis of order
h if every sufficiently large
multiple (resp. greatest c o m m o n
LCM
and
GCD
divisor) of h
bases, nonbases,
and asymptotic
This generates a n e w series of unsolved p r o b l e m s in multi-
theory.
T h e s e can be translated into combinatorial p r o b l e m s
about union and intersection bases for multisets.
Graham,
all n u m b e r s
Lenstra, and Stewart
of the f o r m
[7] have observed that the set consisting of
Z • 3 n, n = 0,1, Z, 3 .... , is a m a x i m a l
nonbasis for the positive integers.
T h e existence of a m a x i m a l
asymptotic
LCM
asymptotic
nonbasis for the square-free integers is still an open problem.
Finally, there is an analogous series of p r o b l e m s about m i n i m a l bases
and m a x i m a l
nonbases for the positive integers under ordinary multiplication.
231
References
i. M.
D e z a and P. ErdBs,
Extension de quelques t h e o r e m e s
densities de series d I elements de
N
sur les
a des series de sous-ensembles
finis de
N,
Discrete Math. 1_Z(1975), 295-308.
2.
Amer.
P. ErdBs and M.
B. Nathanson,
Maximal
asymptotic nonbases,
Proc.
Math. Soc. 4__~8(1975), 57-60.
3.
numbers,
P. ErdBs and M.
Proc. A m e r .
4.
B. Nathanson,
Math.
P. ErdBs and M.
Oscillations of bases for the natural
Soc. 53(1975),
]3. Nathanson,
infinitely oscillating bases and nonbases,
253-258.
Partitions of the natural n u m b e r s
Comment.
into
Math. Helvet. 5__~I(1976), 171-
18Z.
5.
P. ErdBs and M.
in m a x i m a l
6.
nonbases,
B. Nathanson,
N o n b a s e s of density zero not contained
J. L o n d o n Math. Soc. 15(1977).
P. ErdBs and M.
B.
Nathanson,
Bases and nonbases of square-free
integers, preprint.
7.
It. L. G r a h a m ,
H. W.
Lenstra,
Jr., and C. L. Stewart, personal
communication.
8.
Angew.
E. H~rtter, Ein Beitrag zur Theorie der Minimalbasen,
J. Reine
Math. 19___~6(1956),170-Z04.
9.
J. Hennefeld, Asymptotic nonbases not contained in m a x i m a l
asymptotic
nonbas es, preprint.
I0.
number
M.
B. Nathanson,
theory, J. N u m b e r
ii. M.
M i n i m a l bases and m a x i m a l
Theory_6(1974),
B. Nathanson,
s-maximal
nonbases in additive
324-333.
nonbases of density zero, J. L o n d o n Math.
Soc. 15(1977), 29-34.
IZ.
A. StBhr, GelBste und ungelBste F r a g e n ~iber B a s e n der n~turlichen
Zahlenreihe,
13.
Number
J. l~eine A n g e w .
S. Turj~nyi,
Theory9(1977),
Math.
On maximal
271-275.
194(1955), 40-65, 111-140.
asymptotic nonbases of density zero, J.
REMARKS ON MULTIPLICATIVE
Institute
My principal
Atle Selberg
for Advanced Study, Princeton~
We begin by recalling
functions
08540
New Jersey
reason for choosing this rather elementary
attention to the uses of multiplieative
io
FUNCTIONS
topic is to draw
in more than one variable.
the standard definition
of a multiplicative
function of one variable defined on the positive integers:
it is a function satis-
fying the conditions
(i.i)
f(m) f(n) = f(mn)
for
(re,n) = i,
and
(1.2)
f(1) = i.
I have never been very satisfied with this definition~
define a multiplicative
and would prefer to
function as follows:
Write
(1.3)
n = ~
pa,
P
where the product extends over all primes (so that all but a finite number of the
a
are zero).
Let there be defined
negative integers
such that
(1.4)
f (0) = 1
P
p
a function
f (a)
P
on the non-
except for at most finitely many
p.
Then
f(n) = ~ - ~ fp(a)
P
defines a multiplicative
This definition
f(n)
for each
singular if
f(1) = i~
function.
is clearly more general than the previous
f(1) = 0,
we say that
f(n)
otherwise we call
is normal.
*)It should be noted that it permits
f(n)
f(n)
regular.
one *).
If finally
The class of multiplicative
to vanish identically.
We call
functions
233
defined by the standard definition coincides with the class of normal multiplicative
functions according to our new definition°
With the new definition it remains true for instance that if
are multiplicative,
~
f(d)g(~l
\47
dln
it also remains true that if
f((a,n))
f([a,n]) *)
and
g(n)
then so is the convolution
f * g(n)=
then
f(n)
f(n)
,
is multiplicative and
is multiplicative.
a
a positive integer
However, with our new definition,
are also multiplicative,
f(an)
and
something which is not necessarily true with
the standard definition°
Another advantage is that the new definition can be used without change to
define multiplicative functions of several variables.
If we denote by
[n}r
an
(1.5)
r-tuple of positive integers
nl,...,n r
and write
In} r = I I P ~a}r
P
to denote that
ai
n.i = I I p
P
we say that a function
for
i = 1,2,...~r,
f(nl,...,n r) = f([n}r)
is multiplieative if we can write
it in the form
(1.6)
f([n]r ) = ~-~ fp({a]r),
P
where the functions
For each
integers,
p,
if
f({n}r )
satisfy the following conditions.
fp(al,...,ar)
fp(0,...,0) = i
Again, writing
that
f ([a}r)
P
is defined on the
r-tuples of nonnegative
except for at most finitely many
[l}r
is singular if
for the
p.
r-tuple all of whose entries are i, we say
f({l]r) = O,
regular if
f([l]r) ~ O,
and normal
f([l}r) = i.
It is easily seen that if one keeps some of the variables fixed in a multi-
*)We use
[a,n]
to denote the least common multiple of
a
and
n.
234
plicative function one gets a function which is multiplicative in the remaining
variables.
Let us finally mention that in case of functions of one variable the class
of multiplicative functions defined by (1o4) could also be defined by the requirements:
(1.7)
f(m) f(n) = f([m,n]) f((m,n))
for all positive integers
m
and
n.
This is, in spite of its simplicity~ not as
practical as the constructive definition (1.4).
Also one meets complications when
trying to adapt it to the case of several variables°
2.
We shall now concentrate on functions of two positive integral variables~
though as of yet we shall not necessarily assume them to be multiplicative.
that a function
f(m~n)
is symmetric if
f(m~n) = 0
n > m~
and finally normal lower triangular if
all
for
f(m,n) = f(n~m)~
We say
lower triangular if
f(n~n) = i
for
n.
If
t(m,n)
is normal lower triangular and we have two sequences
xm
and
connected by the relations
(2.1)
x
m
=~
t(m,n) Yn
n
then there exists a unique normal lower triangular function
(2.2)
Ym = ~
t*(m~n)
such that
t*(m,n) x .
n
n
t
and
t
are
connected
(2.3)
where
by the
relations
~ t(m,~) t*(~,n) = 6m, n,
6
m~ n
is the Kronecker symbol, or, alternatively we have
(2.3')
~-~ t*(m,~) t(~,n) = 6m, n.
If we assume that
t(m~n)
nlm.
is multiplicative,
t(m,n) = 0
unless
plicative.
Namely~ let us define
it follows immediately that
It is not hard to see then that
~(pr, pS)
for
r > s
t*(m,n)
is also multi-
by the relations
Ym
235
>2~
(2.4)
t(pr pt) ~(pt,pS) = 6
r,s
s<t<r
Constructing
now a multiplicative function
n =
we see that
~(m,n) = [ { ~(pr,pS)
where
m = { { pr ,
P
P
p ,
P
(2.7)
~ t(m,~) ~(~,n) = 6m, n
since the left hand side of (2.7) arises by multiplying together the left hand sides
of (2.4) for all
When
p.
t(m,n)
Thus
~(m,n) = t*(m,n)
which is therefore multiplieative.
is multiplicative as well as normal lower triangular (2.1) and
(2.2) take the forms
(2.8)
x
m
=
t(m,d) Yd
dim
and
(2.8')
Ym =
~ t*(m,d) x ddlm
This generalizes the usual inversion formulae.
We have, of course, also the dual set of formulae:
if
t(m~n)
is not
assumed multiplicative but is normal lower triangular and if the sequences
Ym
xm
and
are connected by the relations
(2.9)
Xn = ~ t(m,n)
Ym'
m
then we have that
(2.9')
Yn = ~
t*(m,n) x mm
Here, since the sums on the right hand side are infinite, one has to assume that,
say, the
Ym
are such that the suma occurring converge absolutely.
instance, is the case if we assume that the
vanish for
(2.10)
then
m
sufficiently large°
xd =
Ym
This, for
(and as a consequence also the
For multiplicative
~ t(m,d) Ym'
dlm
t(m,n)
we get that if
xm )
236
(2.10')
Yd = ~ t*(m,d) x m.
dlm
We call a symmetric function
(2.11)
f(mgn)
Q(x) =
satisfies
Q > 0
positive definite if the quadratic form
~ f(m,n) x m x n
m~ n
for all real sequences
xm
with at least one and at most finitely
many non-zero elements.
For
f(m~n)
positive definite, we can always find functions
normal lower triangular
t(m,n)
(2.12)
g(n)
and
such that
f(m,n) = ~ g(~) t(m,~) t(n,~),
these functions are uniquely determined and can be expressed rationally in terms of
the
f(m,n).
If we, in addition, require
that both
and
g(n)
~(pr, pS)
and
t(m,n)
and
to be multiplieative~ it is easily seen
will also be multiplicative *). Namely we define
~(pr)
by the relations
f(pr pS) = ~ ( p t )
~(pr p t ) 7 ( p S pt) for all
t
and ~(pr pS) = 0 for s > r. For each p this determines
r,s ___ O, t(p r ,pr) = i,
~(pr)
f(m,n)
~(pr, pS)
uniquely for all
r,s > O.
We now construct the multiplicative functions
Writing
m = I I pr
n = I [ pS
we then have
~(m)
and ~{(m,n).
f(m,n) = I If(P r'ps) =
1 1 ~(Pt)
V(pr,pt) ~(pS,p t
= ~(~)~(m,~)t(n,~).
Thus ~
p
t
identical with g and t which are therefore multiplicative.
For multiplicative positive definite
and
t
must be
f(m~n) (2.12) therefore assumes the
form
(2.13)
f(m,n) =
~ g(d) t(m,d) t(n,d).
dlm
dln
3.
conditions
Suppose that we wish to determine the minimum of
xn = 0
for
n > N
and
x I = I.
Writing
f(m,n)
Q
under the side
in the form given by
(2o12) we obtain
2
(3~i)
Q(x) = ~
g(~)I~m t ( m , ~ ) X m l .
*)We assume for simplicity in this argument that f(l,l) (and therefore also g(1))
equals I. This is no restriction since we could otherwise divide by f(l,l) which
is positive.
237
Writing further
(3.2)
Yn = ~ t ( m , n )
m
so that also
Yn = 0
for
n > N,
(3.3)
Xm,
we get
Q(x) =
2
~ g(n)Yn "
n<N
xn = ~
t*(m,n) Ym"
(3.2) gives
(3.4)
m
In particular the condition
(3.5)
xI = I
takes the form
~
t*(m,l) Ym = i.
m<N
The minimum is then by standard procedures found to be
%
(3.6)
.in
t * (m~l) 2
~
'
g(m)
and the minimizing
xn
are given by
(3.7)
Xn = Qmin " ~ t (m,n)* t*(m~l)
m<N
g(m)
Finally, if we assume
t(m,n)
and
t (m,n).
f(m,n)
to be multiplicative,
from the
or not, depending on the nature of the
f(pr, pS)
for
r,s ~ O.
This may be simple
f(pr, pS).
We shall consider an application of the preceding to a sieve problem.
Let us denote by
prl]n that
exact prime divisor of
prI~
and
designate a set of
~(pr)
assume that for fixed
are disjoint if
n
r ~ s.
,p
= i,
we say that
p
is an
n.
Assume now that we have a set of primes
integer
g(m),
These functions have to be determined by computing the
g(pr), t(pr,pS) and t*(pr,p s)
4o
then so are
p
p.
residue classes modulo p
For each
r
the residue classes removed
We introduce the notation
lies in one of the
w(p r)
p
r
with
r > 0
we
to be removed or excluded.
modulo p
nX(p r)
r
and
modulo p
s
to denote that the
residue classes excluded
modulo pr.
We
We
238
define
nX(1)
whenever
for all integers
n,
and further for
nX(d)
if
nX(p r)
prlldo
Let there now be given a set of integers
associated
Xd = 0
d > i,
a weight
for
Wn ~ O.
d > Z
We assume that
and leaving the other
n
W =~w
%d
with each of which there is
n < =o.
Writing
k I = i;
as free real variables,
we form the
expression
(4.1)
Q(%) = ~ w n
n
Clearly
Q(%)
~
.
)
is always an upper bound for the sum of the weights
which remain after we have removed those that lie in any of the
residue classes modulo each
weights
wn
the quadratic
pr.
form
of the integers
w(p r)
Under rather general assumptions
Q(~)
excluded
about the set of
can be written in the form
Q(X) = QI(X) + R
where
Ql(k) = W
and
f(d,d')
and
R
I%dl.
f(d,d') kd kd'
is a symmetric multiplicative
is a remainder
The machinery
function (positive definite,
term generally bounded by a simple quadratic
from the previous
subject to the side conditions
determined
~
d,d'
by the requirement
on the
that
section then applies,
%'s,
R
of course)
form in the
one can minimize
the choice of the parameter
Z
QI(~)
is then
should be small enough not to spoil the
result.
We shall apply this technique
(so that we assign the values
~
= i~
introduce a symmetric multiplieative
if
r = s
d'
are compatible
now get
or if
rs = O,
if
to the case of an interval
for
n 6 Ix
function
otherwise we define
E(d,d') = 11
otherwise
and
~(d,d')
I
Wn = 0
x
outside
by defining
E(pr,p s) = O.
of length
I x.
X
We
E(pr~p s) = 1
We say that
d
they are said to be incompatible°
and
We
239
Q(~) =
E
nEl x
k
< x E
f(d,d') ~d ld'
d,d'
)
(4°2)
+ E
IXHI
{~d' [ w(d,d')
~(d,d'),
d~d'
where
(4.3)
f(d,d') = w([d~d'I)
[d,d']
Here
w(d)
is the multiplicative
~(d,d').
function defined by
w(d) = I I
r
00(i) = i
and
W(P r)°
p lld
An alternate form of the upper bound for the interval
follows *) .
w(u) > I
Consider a function
for
u
in
I
--
and
w(u)
can be obtained as
defined on the real line~ such that
w(u) > 0
x
Ix
always.
We furthermore require that its
--
transform
fourier
W(V) = 7 W(U) e 2~iuv du
.oo
should vanish identically
for
i
Iv I >--~ .
We then have
z ~
(4.4)
Q(%) _ < ~
_ao
w(n)~
%
LnX(d)
It can be shown that we can choose
w(0) < X + Z 2.
d•2
= wA(O)
w(u)
E
d,d'
f(d,d') k d ~d' "
satisfying our conditions and such that
Thus we get
(4.4')
Q(%) ~ (X+Z 2)
~
f(d,d') %d %d'"
dgd'
To use the results of the previous section to minimize the quadratic form on the
right hand side of (4.4')~ we observe that we have
and
f(pr,pr) = f(pr, l ) = f(l,pr) = ~ ( r )
Writing
P
(4.5)
we have
8(p r) = 1 -
g(1) = i,
*)See Selberg [i]o
and for
r > 1
~
~(ps)
s
l<s<r p
f(pr,pS) = 0
for
(r-s)rs ~ 0,
240
g(pr) = (e(pr-l) . @(pr)) 8(pr)
6(pr-l) "
(4.6)
For
r > s~
we find
e(pr-l)_@(pr),
if
s = 0
e(pr'l)'e(pr),
@(pS)
if
0 < s < r .
t(pr'pS) = I "
(4.7)
Similarly for
r > s
8(pr-l)-@(p r)
*. r s.
Jt (p ,p) =
(4.8)
1
@(pr-l)
'
8(pr-l).~(p r)
6(pr-l)
'
if
s = 0
if
O<s<r.
Thus
*r (
t (p ~i) 2
g(pr)
(4.9)
i
6(pr)
i
£(pr-l)
1
Thus the minimum of
f(d,d') Xd Xd'
is
i
d<Z prHd
e(p r)
e(p ~'I)
From (4.4') we thus get the upper bound of the number of integers that the sieve
leaves in
I
as
X
X+Z
(4.10)
2
prlld e<p r)
e<
A result~ identical in form with (4.10)~ has been given earlier by Gallagher
and Johnsen
*)
, who derived it from the large sieve and under the rather restrictive
assumption that for each
*)See Gallagher [2].
r > i
the
m(pr)
residue classes excluded modulo p r
are
241
equidistributed among the
pr-I @(pr-l)
residue classes that remain modulo pr-I
after the earlier exclusions modulo p,o..,p
instance~ that
P
P o @(pr)
e(p r'l)
w(Pr)
r-i
i)
9(pr-
r-i
This restriction implies, for
is always an integer (or~ otherwise expressed 9
is always an integer) *) .
References
Io
Ao Selberg:
Remarks on sieves°
Proceedings of the 1972 Number Theory
Conference~ University of Colorado~ 1972~ pp. 205-216.
2o
P. Xo Gallagher:
Sieving by prime powers°
Proceedings of the 1972 Number
Theory Conferenc% University of Colorado~ 1972~ pp. 95-99°
*)When I first saw Gallagher's paper [2], I thought that the unnecessary restriction
was due to faulty technique in the use of the large sieve inequality. As I~
however~ was unable to derive (4.10) from the large sieve without imposing the
same restriction, it remains an open question whether the more general result
given here is implied by the large sieve inequality or not.