Internat. J. Math. & Math. Sci.
Voi. 8 No. 2 (1985) 283-302
283
EXTENSIONS OF SOME FORMULAE OF A. SELBERG
CLAUDIA A. SPIRO
Department of Mathematics, State University of New York
Buffalo, New York 14214 U.S.A.
(Received March 3, 1984)
ABSTRACT.
This paper is concerned with estimating the number of positive integers up
to some bound (which tends to infinity), such that they have a fixed number of prime
We obtain estimates which are
divisors, and lie in a given arithmetic progression.
uniform in the number of prime divisors, and at the same time, in the modulus of the
These estimates take the form of a fixed but arbitrary number
arithmetic progression.
of main terms, followed by an error term.
KEY WORDS AND PHRASES.
prime divisors.
Extensions
of
some formulae,
1980 MATHEMATICS SUBJECT CLASSIFICATION CODE.
i.
formulae of A. C_’b’$
10L20, 10H20.
INTRODUCTION.
Let x > 3, choose B > 0, and let j
B log logx be a positive integer. In
1954, A. Selberg (cf. [i], Theorem 3) showed, for example, that the number of positive
integers not exceeding
x
which have exactly
i
(j-l) !logx
uniformly in
j,
where
kdzJ-I
(z)
(z) (log x) z
distinct prime divisors is
j
)I
z=O
+ 0B
is a certain entire function.
(i I)
(logx)2
We replace
(z)(logx) z
by an expression of the form
M
Z (log x)
and replace the error te by
z-
(1.2)
z
0N(X(loglogx)J(logx)-M/jl)
Moreover, we obtain
results of this strength for, e.g., the number of positive integers not ceeding
ich lie in an arithmetic progression (say,
have precisely
unlfo in
k
j
distinct
pre
not exceeding a fed power of log x,
obtained comparable results for fed
for
M
ere (h,k)
eorem i, below).
k.
I)
h modk,
divisors (see
and in
H. Delange [2]
j
G.J. Rieger [3] obtained related results
account of the history of the problem, see Section 4 of Norton,
For a greater
[5].
Let
In Splro [6], we apply our results to prove the following result:
denote the number of positive integers dividing the positive integer
be any positive integer. Then there are computable constants R I,
R > 0, such that
1
M
d(n)
divides
n+l]
=x
=i
and ich
Our results are
1, ich we will discuss in greater detail in Section 7.
#In&x:
x
R(logx)
-
+
,,
n,
0s(X(logx)-M-%)
d(n)
and let
with
M
(1.3)
C.A. SPIRO
284
2.
THE PLAN OF THE PROOF.
NOTATION.
Throughout this paper
integers, and
and
s
p
denotes a prime,
k, n,
and
N
represent positive
We will let
stand for complex numbers.
z
denote the real
g(x) and f(x)
O(g(x)) will have their usual
f(x)
and
we
write
will also
f(x) < g(x) to mean that f(x)
O(g(x))
meanings,
By
0
either f(x)
or f(x) <
we will signify that
(g(x))
...g(x)
a,b
a,b
where the implied constant possibly depends on a,b
f(x)
O(g(x))
Similarly,
part of
The expressions
s
[a,b
we will write
]-sufficiently large
will possibly depend upon
to indicate that how large is sufficient
When only one variable is inside the curly
a,b
A product or sum
brackets, we will omit them, and write, e.g., a-sufficiently large.
of the form
H
or
P
P
example,
pn(l. + I/p)
dividing
n
function of
n, write
function by
[x]
we put
F(z)
p
x,
(n)
i
+ i/p,
taken over all primes
nx
(n)
for the Euler phi-function of
denote the Riemann zeta-function by
designate the Dirichlet L-function of s and
log log x
and
log log x
L3x
Finally,
denote the
n,
(s)
Co,Cl,...
use
n(x)
for
and the gamma-
L(s,x)
by
for the greatest integer not exceeding the real number
L2x
p
is assumed to extend over all
As is standard, we let
x
not exceeding
n
the number of primes
write
denotes the product of
Similarly, a sum of the form
positive integers
Mbius
respectively denotes a product or sum over primes; thus, for
and
For simplicity,
x
denote positive
absolute constants.
A > 0
Let
be a nonnegatlve integer-valued additive
be constant, and let
function such that
(n)
Alogn
for all
n,
(2.1)
(p)
I
for all
p.
(2.2)
where (n)
(n) and (n)
(n) is the number of primes
Examples of functions satisfying (2.1) and (2.2) are
is the number of distinct primes dividing
n,
dividing
by
let
wll
and
n,
counted with the multiplicities with which they occur.
the arithmetic function which takes the value
(n)l(n)l
denote an arithmetic function which is either
character
,
and for complex numbers
and
s
z
or
II
Also, designate
at
n.
set
E X(pm) z(pm) p -ms
f(x,z,x;)
We will
For any Dirlchlet
(2.3)
p m=0
E(s,z,x;I)
g(s,z,x;)
+ X(p)zp-S)
11 (I
P
<p
g(s,z,x;ll)
x(pm)
m=O
H(I +
(2.4)
z(pm)p-mS)(l_x(p)p-S)Z
X(p)zp-S)(l-x(p)p-S) z
(2.5)
(2.6)
P
wherever these products converge.
cause no confusion.
(i
X(P)P
--S Z
Since
II
is not additive this notation should
By the complete multiplicativity of
X,
we can expand the factor
by the Binomial Theorem and formally write
f(s z,x;)
7
n=l
a
Z
(n;)x(n)n
-s
(2.7)
285
EXTENSIONS OF SOME FORMULAE OF A. SELBERG
7.
g(s,z,x;)
b (n;)x(n)n
z
n=l
a
where the coefficients
z(n;)
and
b
z(n,)
s
(2.8)
are independent of
X
first, we will
The plan of the proof will be to argue in the following manner:
show that the derivation of Hilfsatz 5 of Rieger [3] can be used to estimate the sum
Z
nx
Az(x,x;)
(2.9)
az(n;)x(n)
We will establish Lemmas 3 and 4 of the next section for this purpose.
[4].
c
O
X
is
X0
The estimate made in Rieger [5] in this case
principal, more effort will be required.
relies on Hilfsatz 13 of Rieger
If
This Hilfsatz implies that there is a constant
> 0 such that
i- p
elk
Is-i[
for
log(2k)
c
-s
i
l-p
O([s- l[L2(4k))
+
In our Lemma 2, we replace the right side by an asymptotic
o
and this expansion leads to an asymptotic expansion
expansion in powers of (s-i)
-i
descending by powers of log x (see Lemma 7, below). After we
for x A
estimate
z(X’X0;)
Az(X,X;
we argue in a manner similar to that of the proof of Theorem 3
of Selberg [6], to obtain our main result.
We remark that we can obtain comparable estimates to the estimates which we
A z (x,x;) (see Lemma 7 below) for a wider class of arithmetic
functions a z(n;) than the class given by (2.7), (2.3), (2.4), and our restrictions
obtain for the sum
We restrict ourselves to these examples here for simplicity of exposition.
on
3.
PRELIMINARY RESULTS.
LEMMA i:
7.
i)
p
-i
(logp)
n
n
p[k
7.
ii)
p
-i
(i-
plk
REMARK.
(2.13)).
(k)
< L 2(3k)
(3 3)
The lemma is trivial if
n(x)
I
n
is done in Rieger
[4] (see equation
1
Otherwise, let
primes dividing
k,
q
denote the
so that
q
(k)
th
is at
Therefore, it follows from the Chebyschev bounds
prime.
that
q
-i
k
(log k)/log 2]
(log k)/log 2]
most the
p
(3.1)
i.
(3.2)
The special case of i) when
There are at most
th
prime.
Since
n
if
The estimates ii) and iii) are well known.
PROOF.
for
k
p-l)-i
n
+ CI
L3(8k)
[k
ili)
(e2(3k))
<
(logk)
L2(3K
(log p )n is a decreasing function of
p
(3.4)
if
p
is sufficiently large, we
can conclude that
7.
pl k
p
-i
(log p) n
7.
n
pq
p
-i
(log p) n
286
C.A. SPIRO
Hence, it follows from (3.4) and the Chebyschev estimate for
.
p
-i
plk
(logp)
n
(logq)
n
n
n
that
(L2(3k))n
If we repeat the argument with
This inequality gives i).
u(x)
(3.5)
0,
n
and replace our
second application of the Chebyschev estimate by the estimate
+ 0(I)
Exponentiatlng ii) yields ill). //
[s-l[log p < i for every prime
we obtain il).
LEMMA 2.
that
L 2(3q)
P
Pq
Assume that
[s-l[e2(3k)
C,
N-I
l_p-S
Plk
1-p
i
-1
+
are real-valued, depend only on
n
C.
and
Then
(s-llnPn(k) + 0N, C (([s-l[L2(3k)) N)
Y.
n=l
P (k)
where the coefficients
k,
dividing
for some fixed but arbitrary positive constant
N > i, we have
for every fixed integer
H
p
k
and
(3.6)
n,
and
satisfy
P (k)
0
n
PROOF.
n
From the Taylor expansion for
((L2(3k))n)
u
(3.7)
about
e
0, and our first hypothesis,
u
we can deduce that
l-p
l-p
-s
-I+ i
-i
e
(l-s)log p
p-i
j=l
+ E(s,p,N)
J!
(3.8)
where
0(([s-I logp)N p-l)
E(S,p,N)
(3.9)
and where
1
-I
NI
(log p) J (1- s) j
j!
+
E(s,p,N)[
7
(3.10)
IT
By (3.9) and (3.10), we can take the logarithm of the right side of (3.8), and obtain
log 1-p
-sN-I
l_p-1
7.
(s-i )m (logp)
m=l
em
pk
l-p
-s
-I
exp
T
p k
m=l
where
term in
Qm(U)
(3.12)
log
p
Z:= I emu-I
is
7.
l_p
+
ON
<([s-l[lo p)N
8
(3.11)
P
-s
-I
(3.12)
(l!PI) Qm((P_l)-l) + ON
([s-lllogp)
P
N
P
It now follows from Lemma i, part i) that the error
0N((Is-IL2(Bk))N)
clude from this Lema that
em( p_ i)-
Accordingly,
l-p
(s-l) m
exp
m
=i
for appropriate constants
l-p
m
Since
Qm
is a polynomial, we can also con-
EXTENSIONS OF SOME FORMULAE OF A. SELBERG
_(logp)m__
y.
m
((p i)-I
p-i
plk
287
(e3(3k)
(3.13)
If we combine this result with our second hypothesis, we find that the expression in
(3.12) in curly brackets is
of
eu
about
follows.
0
u
ON,C(1)
Consequently, we can apply the Taylor expansion
to the right side of (3.12), and the first equation of our lemma
To obtain the estimate for the coefficients in
R (k)
coefficient
of
m
and that the coefficient
Rml(k)Rm2(k)...Rm.(k)
LEMMA 3.
. Ibz(n;
m
(3.12)
in
Pn(k)
+
i
l >5
Pn(k)
we observe that the
is the quantity we estimated in (3 13)
is a finite linear combination of terms of the form
with
1 >
If
(s-i)
+ m.j
//
n
is fixed, then there is a real number
B=
B(Ol, )
> I
such that the following statements hold:
i)
)x(n)n
-s
(3.14)
n=l
converges uniformly in
ii)
for every
s, z,
o
2
and
with
X
o >
Izl
and
i
B
> I,
" laz (n; )x(n)n
-s
(3.15)
n=l
converges uniformly in
,
s, z,
and
with
X
o >
,
If
l
take any
B <
REMARK.
B,
then these results are valid for
1
B
It follows from (2.1) and (2.2) with
n
exp(o I
Consequently, we have
Izl
and
2’
these results hold regardless of the choice of
<
and
B.
B
2Ol
II
If
or
does not depend on
II
For
we can
)/A]
exp[(c I
Cl -1/2
2
1
-)/A]
2
p
that
(2)Alog 2
i
so that the result for
f
is
superior to the result for an arbitrary choice of
PROOF.
II"
First, assume that
Formally, it follows from the Binomial
Theorem and (2.2) that
(pm) p -ms
(l- p -s)z Y- z
m=O
I-
--+Zzs
+
=2
p
for every prime
b (n;)
z
p
-
(3.16)
(-I)(Z)
(_i)( z p-S
+
z
(pm) p -ms
m=2
Hence, from (3.16) and the definition of the coefficients
we can conclude that
j=O
Ib z (pJ-)p-S
i
+ z
2
I()I
0
Denote the sum on the left of (3.17) by
Let
7.
:0
B
be strictly between
we deduce from
I
(2.1) and (3.17) that
and
+
i
+ 2 +
p
(3.17)
m:2
E(p,z,s;)
exp{(o I
i
-)/A}
For
zl
B
and
o
I
C.A. SPIRO
288
E(p,z,s;)
+
i
+
B2p -2i
(!
I(z )IP
0
zl
No. for
E(p,z,s;)
term in
o
=2
)i
" P
m=2
+
(3.18)
m(AlgB-Cl)
([B] + i)
-
+ (1 + Bp
&
Now
I -AlogB
o
0B(P -203)
(3.20) is
jr
I)LB
( +
(3.19)
i=l
(3.18) and (3.19) yield
3
-O1
z
I()IP
i)
in (3.18) is uniformly bounded in
Hence, the last sum on
o
-o
-
B,
l(z )i
where
+ (l + Bp
OB(’BP- 1)+OB( P-2
-o1)
z,
so that
TM
E
=2
>i
3
and
p
(3.20)
m=2
B, so that the final error
by our choice of
Moreover,
Bp
-2
I
=2
-20
p
i
( + 2) B 2
7.
-o
1
(3.21)
=0
It follows from (3.20) and this last inequality that
1 +
E(p,z,s;)
OB( p
-201
-2o
+
0B( p
o
In view of our notation, (3.22), and the fact that
o
and
I
-
2
i
both exceed
3
conclude that the product
7.
p j=O
converges uniformly in
and
s, z,
Ibz(p j ;)x(pJ)p-Sl
For fixed
B,
there is a number
o
4
E
o
with
X,
series in (3.14) is uniformly bounded in
i
(5, l)
Izl
and
ol,
s, z, and X,
with
B.
o > o
B <
such that
I
with
o >
0
4
z
and
n>x
B
o
But if
Ib z (n;)x(n)n-Sl
Y-
<
n>x
and
I
4-1
-
n=l
uniformly in
x
x, z,
verges uniformly in
i
(3.23)
Izi
and
B.
i
exp[(o4- )/A]
s, z, and
then we have
Ib z (n’)In-04x@4-01
x
Consequently, as
x
we
Accordingly, the
what we have just proved, the quantity in (3.14) is uniformly bounded in
X,
(3.22)
(3.24)
__[bz(n;)ln-04
tends to infinity, the sum on the left of (3.24) tends to zero,
and
s, z,
X
with
and
o
X,
o
I
and
Izl
as asserted.
B.
So, the sum in (3.14) con-
To obtain the uniform convergences
of (3.15), in view of the derivation of (2.7) it is enough to show that
y:.
[Xcpm)z(pm)p-ms[
p m=O
is uniformly convergent.
By (2.1) and (2.2), we have
(3.25)
By
EXTENSIONS OF SOME FORMULAE OF A. SELBERG
. Ix(pm)z(pm)p-mS
< i
+ p-O +
7.
o
o
3
A log B >
I
,
i
A log B)
(3.26)
m=2
m=O
By hypothesis,
p-m(o
289
and consequently
IX(pm) z(pTM) p-mS
<
I +
p-O
-20
+ P
-n
3
2
3
m= 0
m= 0
+
i
p-O
(3.27)
-2o
+
0B,Ol(
p
3
Accordingly, the quantity in (3.25) is uniformly convergent, as desired, and the lemma
is proved in this case.
,
In the event that
be derived from (3.17).
the uniform convergence of the quantity in (3.14) can
In this case, the uniform convergence of the quantity in
(pm)
(3.25), and hence, of the quantity in (3.15), follows from the fact that
whenever
1
m
i
These derivations are similar to the argument that we have just
made, and will therefore be left to the reader.
into (3.17), the last sum in (3.17) is the geometric series
If we put
=21 zlmp,p
in
If we fix
z,
n
for
s,
and
o
B 6 (1,2
o > o
with
I
zl
& B
-201
OB( p
then this series is
and
,
1
z
-
II
Finally, suppose that
(3.16), for every prime
given above,
As in the deri-
we have
p
+ z
uniformly
The rest of the proof of the result
is similar to the proof of the result for the case
and will be left to the reader.
vation of
I)
i
z
+
+
z
z
(2,)(-i)
2
p
-s
p
(3.28)
-ml
(3.29)
Hence, in view of our notation, we find that
j=O
]b z (pJ;ll)p-Sl
1+
lzl2p -2I
-l
=2
Again, the remainder of the proof in this case is similar to the proof for the case
I[
in which
//
and we omit the details.
For the remainder of this paper, we will assume that
that
B
B(Ol,)
we will take
o
>
to be fixed.
I
only inasmuch as it depends on
the dependence upon
LEMMA 4:
depending on
uniformly in
PROOF:
For
o
I
i)
satisfies conclusions
i
o
i
I
>5
of Lemma 3.
ii)
is given, and
Furthermore,
If a constant implied in an error term depends on
B, we will indicate the dependence on
B,
o
1
but not
I
x
i
< y and for
B
such that
E
x<n<y
az(n;)x(n)
and
and
I > o
z[
<
B,
0B,o I ((y
there is an
x) (log y
B
> O, possibly
+ y i-
(3.30)
z
Since
f(s,z,x;)
g(s,z,x;)((s))
z
(3.31)
290
C.A. SPIRO
[3] that
it follows from (2.7), (2.8), and equation (i) of Rieger
a
z
n. )’
(n’{)= 7. d z ()b z (’
-
where
(3.32)
n
n=l
dz(n)n
((s))
(3.33)
Idz(n)
We can conclude from equation (2) of Rieger [3] that
n
and
z
with
z
B,
,
d[B]+l (n)
for all
so that (3.32)yields
laz(n;{)
d
n
[B]+I
(n)
lbz(n;g)]
(3.34)
We claim that it suffices to show that
. n d[B]+l(n)Ibz(n;)l= .
[B]
7.
nx
(oI,B)
for a constant
in
z
with
j=l
Izl
> O, and
j,B,z
constants
x(logx)J+OB ’i (x I-
(3.35)
which are uniformly bounded
j,B,z
Indeed, from (3.34) and (3.35), we will be able to deduce
B.
that
a
z
(n"(n)I’
)X
&
’,
7.
(n) b z (n; )
d
n B +I
[B]
j=07. J’B’z[Y(lg y)J
y< n& x
(3.36)
x(logx)
j]+OB,Ol(yl-
The Mean Value Theorem implies that
y(logy)J
x(logx) j
0j
(y
x) (log x) j-I
for
j
i,
(3.37)
According to (3.34), the
and combining this fact with (3.36) verifies the claim.
left side of (3.35) is
Z=
7.
nx
[bz(n;)l x/n dz () ZI +7.11+7.111
(3.38)
where
7.1
(n;
x lbz )[
7.
n
7.
7.11 x< n& x/2
7.111
Z
x/2<n&x
d
& x/n
(3.39)
z
z
& x/n
IbzCn;)
(3.40)
(3.41)
By equation (12.1.4) on p. 263 of Titchmarsh [7], we have
[B]
7.
x d[B]+
for certain constants
I()
Z k
x(logx) j +
j,B
j=0
kj, B
Substituting (3.42) into (3.41)
yields
i
OB(X
i
[B]+I (log x) B)
(3.42)
EXTENSIONS OF SOME FORMULAE OF A. SELBERG
EII
(log x) [B]
x
x< n&x/2 Ibz(n; g)I
E
B
291
(n; ) ,- Xn (log x)
x< n.& x/2 bz
B]( )i l
(3.43)
Hence, we can deduce from Lemma 3, part i) that
0
Eli
B(x
(i +c;i)/2
(logx) [B]
(3.44)
Part i) of Lemma 3 also implies that
EIII
7"
x/2<nx
X
..bz(n;)(
<
c
I
x
(3.45)
B
Finally, combining (3.42) with (3.39) gives
7"l=
7"
n
Ib Z (n;)
x
n
[B]
E
j
j=O
j
x(logX
B
+
To bound the error term, we note that if
OB nx bz (n ) [(
1
7.
1
B--$]zo I,
X
1
B+2
(3.46)
then part i) of Lemma 3
yields
7"
nNx
B+2
]b z (n; )I( X
0B
(3.47)
Otherwise, we can apply Lemma 3, part i) to obtain
7.
)
n [bz(n; [(
x
i
i
1
I
B+2
()Cl-l+B+--
x
7.
n [bz(n;) [(
0
(3.48)
12B--+
x
B
/.
It remains to estimate the main term in (3.44).
x j
(log x) j
(log)
If we write
(i io$
o I/
n
j
(3.49)
and apply the Binomial Theorem, we find that
[B]
7" k
j=O
j B
[B]
j
x(log x
7.
j,B
j=O
If we make the change of variable
[B]
u
x(logx)
m,
j
j
(3.50)
we can rewrite this expression as
[B]
7. x(log x )u 7. k
(-I) j-u
j,B
u=O
j=u
j
u
(log n) j-u
(3.51)
Hence, the main term of (3.46) is
[B]
7. x(log x)
u=O
u
We want to write the sum on
[B]
7.
j=u
n
kj,B
(-log
j
u
n)J-U[b z(n, )
(3.52)
n
as the sum from
n
i
to
,
plus an error term.
292
C.A. SPIRO
It follows from part i) of Lemma 3 that this remainder term is
E
nz
0
’bz(n;)
x
l
n
)
n
u
(log n) j- /n
since the factor
J-u
(llOg_ ln)
l-Ul
<
OB, J’Ul
(logx) j-ux
is a decreasing function of
(l-l)/2)
(3.53)
Hence, we can
n.
combine whichever one of (3.47) or (3.48) applies to the error term in
(3.46) with
(3.38), (3.44), (3.45), (3.52), and (3.53), to obtain
[B]
7
7.
x(log x)
[B]
u
k
51.
u=0
j=u
for some constant
the sum on
E
The claim now follows with
(3 54)
equal to
j ,B,z
in (3.54), since part i) of Lemma 3 implies that
j
(log
n)J-Ulbz(n; )
is uniformly bounded in
z
zl
for
(3.55)
n
n=l
4.
+OB,; I (xl_
n
n=l
> O.
(uI,B)
g
n)J-Ulbz(n;)
(-log
j
u
j,B
<
//
B
THE ESTIMATE FOR A z(x,X;)
Let
c
be the constant
2
< 6 <
0
assume that
c
from Rieger [3], put
6
C(Ul)(log(Sk))-i
Then, we will define the contour
the negative real axis.
the following three paths:
min[c2,l-Ul]
c(o I)
0
Cut the complex plane from
S(k,)
the straight line segment from
i
to be the union of
C(Ol)(log(Sk))
along the negative real axis below the cut; the circle of radius
i
i, traversed once counterclockwise, beginning and ending at the point
not including this point); the straight line segment from
and
along
to
I
to
-I
to
and center
(but
i
l-C(Ol)(log(Sk)
along the negative real axis above the cut.
LEMMA 5.
Let
k
< e lgx
results are uniform in
z
and
and let
k
with
X
be a character modulo k
Izl
-c
3
If
X
is nonreal, then
ii)
If
X
is real but nonprincipal, then we have
-c
where
0B(xe
Az(X,X;)
i)
Az(X,X;)
3/log x
+
log x
(4.1)
xB(k
OB ((k)--(lg 2k) c4
B
(4.2)
max[o:L(o,k =0]
8(k)
iii)
OB(Xe
The following
B:
If
X
X0
is principal, then we have
-c
Az(X,X;)
Lz(X,k)
+
OB(Xe
/
3 log x
(4.3)
where
L (x,k)
z
PROOF.
i
s
f(x,Z,Xo;
S(k,6)
x__ as
s
As we indicated at the end of Section 2, we
replace
(4 4)
(L(s,x)) z by
-i
EXTENSIONS OF SOME FORMULAE OF A. SELBERG
g(s,z,X;)(L(s,X)) z
f(s,z,X;)
in the proof of Hilfsatz 5 of Rieger
g(s,z,X;)
293
[3].
To insure that Lemma 3 is applicable to
on the paths of integration occurring in the proof of Hilfsatz 5, we
replace the constant
c
in the equation following (6) of Rieger [3] by our constant
6
Now part i) of Lemma 3 implies that g(s,z,x;) is uniformly bounded in
and
B, so that g(s,z,x;)=OB(1) on each of these paths of
I)
c(o
o
o
Izl
I
Hence (4.4)
integration.
yields
If(s’z’x;)l
B
i(e(s’x) Izl
(4.6)
Furthermore in place of the estimate
d (n)X(n)
OB((y- x)
Z
x<ny
(log y)B
uniformly for
Izl
have Lemma 4.
We leave the details to the reader.
n
Q(k)
Set
B
I
and for
x
< y (see
+ Y
I
B+2
(4 7)
line 19 on p. 185 of Rieger
[3]),
we
//
p.
Plk
LEMMA 6.
let
Choose
> 0, assume that k < exp]logx and that Q(k)
u
be a nonprincipal character modulo
X
and suppose that
k
z
B
(logx)U
Then
-c ] log x
A (x,X;)
z
uniformly in
PROOF.
and
k
0
Bu
(xe
3
(4 8)
z
is nonreal, the result follows from Lemma 5, part i), afortiori.
If
X
real then we can conclude from Hilfsatz i of Walfisz [8] that X is also
a character modulo Q, for at least one element Q of [Q(k),2Q(k),4Q(K)]
According to a theorem of C. L. Siegel (cf. Estermann [2], Theorem 48), if e > 0 is
(n/4)
given and n is e-sufficiently large then 8(n) is less than I
If
X
is
-
Consequently if Q(k) is -sufficiently large we can deduce from Lemma 5, part ii),
that
k replaced by Q
with
than i,
Select
and suppose that
N
u
zl
> 0, assume that k < exp iogx
B
Let
(4.9)
Now there are only
X0
N
(X,Xo;)
x 7.
=i
k, N,
and
Q(k)
(log x)
k,
u,
and
Then we have
z(k) (log x)Z+
uniformly in
and that
denote the principal character modulo
be a fixed but arbitrary positive integer.
Az
-u (log x) C4B
for which Q(k)
and (4Q(k)) are all less
(2Q(k))
In each such case (Q(k))
so that our lemma also follows from part il) of Lemma 5. //
LEMMA 7.
let
l-(log x)
fails to be C-sufficiently large for
k
finitely many values of
i/(3u)
x
to obtain the desired result.
i/(3u)
Thus, we can take
-c3 log x +
0B(xe
Az(X,X;)
z
0B,N,u(X(logx)Re z-N-I(L2(3k))N k) )Re z)
where
(4.10)
294
C. A. SPIRO
%z(k)
(-i)
!r(z-+l)
(4.11)
ks----i
s--i
Furthermore,
IE(zuniformly in
k
+
l)z(k)l ,B (L2 (3k))
(4.12)
z
and
First, we rewrite (4.4) as
PROOF.
X
L z (x,k)
2 S(k,6) x
S-i
(s-l)
{ (s-l)zf(s’z’xO;)!
-Z
s
(4.13)
ds
By (4.5),
(s-1)Zf(s,z,)O;)
(4.14)
where
h(s,z k;)
s
-I
Z
g(s,z,x0;g) ((s
(4.15)
1) (s))
We can conclude from (3.8), (3.10), and Lemma 2 that
l-p
plk
l-p
N- i
-s
i=
-I
n
I
(s-l)nPn(k) + 0N((Is- llL2(3k)) N)
7/10, where
has magnitude less than
Pn(k)
depends only on
k
and
(4.16)
n
On((L
2(3k))n)
(4.17)
Hence, the Binomial Theorem implies that
Z
plk
l-p
-I
j
J
0
E
n;1
(s-l)nPn(k) + 0n ((Is-llL2
+
N-I
=I+
j=l
uniformly in
z
and
k
Rjz(k)
Now
(s-l)(s)
j
0B,N(IS- IIN(L2(3k)) N)
R. (Q)(s-1) j +
3z
where the coefficients
(3k))N+l))
(4.18)
0B,N((Is-llL2(3k)) N)
Rjz(k
of
(s- i) j
satisfy
OB,j((L2(3k))J)
is analytic and bounded away from zero on
(4.19)
Is: Is-11 =1/2].
Further-
more, part i) of Lemma 3 implies that
z
with
that
Izl
Izl
B
B
and
o
Hence, it
g(s,z,X;)
OB(1) uniformly in k, s, and
and that g(s,z,X;) is analytic in
>
l
l provided
follows from (4.15) and Cauchy’s inequality for the
coefficients of a power series that
h(s,z,k;)
7.
j=O
T
z,j
(k)
(s-l) j
(4.20)
EXTENSIONS OF SOME FORMULAE OF A. SELBERG
T
with coefficients
z,j
(k)
295
which satisfy
IT z,j (k) <B ,o I
pl k
Thus,
j=N
B’I
z, j
uniformly in
s
any point
j=N
zl
with
z
and
k, s,
I-O1 )
B
and
along the path of integration
B’OI
Is- i
S(k,$)
(i-o I)
Is-ll
satisfies
so that
N-I
(k))
j=O rz,j(k)(s-l,J + OB,Ol (s-- ii N+Ik----
h(s,z,k;)
along this path.
i
Is- iI N
(4.22)
By construction,
i
5(1-Ol)
Re z
(4.23)
So, we can conclude from (4.14), (4.18), (4.19), (4.21), and (4.23)
that
(s-l)Zf (s, Z,Xo; )
N-I
s
.
(4.24)
j=0
where the coefficients
3z
(k)
satisfy
(4.25)
Since power series representations are unique, we must have
(4.26)
We can conclude from (4.4) and (4.24) that
.
N-I
L (x k)
z
a. (k)xl(x,z-j k) +
3z
j=O
+
OB,o I,N
(x (L (3Q))
2
(---)Re z
N #(k)
S(k,6)
[(s- I)N-z[ O-l[dsl)
(4.27)
where
i
-i S(k,6) (s- l)-WxS-lds
l(x,w,k)
Let
C(b)
denote the contour which goes from
axis below the cut, traverses
real axis, above the cut.
w
and
k
lwl
C(Ol)
and then returns to
along the negative real
along the negative
/logx,
k <
SC(b (s-l)-WxS-lds + 0B,OI,N
l(x,w,k)
uniformly in
S(k,)
Since
to
(4.28)
B + N
(4.29)
-(s-l)logx= u, and recall
the Hankel integral representation for the gamma-functlon (cf. the last equation on
p. 245 of Whittacker and Watson [i0]), we find that
with
If we let
C. A. SPIRO
296
(logx)W-l+ OB,o le
I’N
l(x,w,k)
To bound the remainder term in (4.27), we separate
K(6)
6
and the union H(k,)
-i
Then
(logx)
(4.30)
S(k,6)
into the circular part
of the two horizontal strips of
7k(6) }(s- l)N-Z[xO- llds[
<
(logx)
Now, it certainly suffices to prove this lemma when
N
S(k,6)
Choose
Re z-N-i
> B
(4.31)
In that case, we have
(4.32)
Upon making the change of variable
,[
(sH(6)
(i-o) log x, we arrive at
Re z-N-I
Re z) (log x)
t
l)N-ZxO-ldsl B,NII(N
B,N(IOg x)
Re z-N-i
(4.33)
Thus, the error term in (4.27) is
OB,oI,N
((L 2(3Q))
N (k)
Q--)
Re z
x(log x)
Re z-N-i
(4.34)
Furthermore, by (4.15), the main term in (4.27) is
N-IE F(z-j)l jz(k)x(lgx)Z-J + 0B,oI,N (xe-c5lgx
(4.35)
j=O
The lemma now follows from (4.26), (4.27), and part ill) of Lemma 5. //
5.
THE STATEMENT AND PROOF OF THE MAIN RESULT
Let
h, j, and
k
j z O, k m i,
be integers with
#In
.(x,k,h;)
x:(n)
THEOREM i.
Let
Q(k)
Choose
be integers
h, j, k
(log x)
u
u
x
(k) (j-)
+
uniformly in
PROOF.
h, j,
and
-
x(n)
x,(n)
i
Set
(5.1)
(5.2)
j
> 0, and let N be a fixed but arbitrary positive integer.
i
and
i
k < exp]lgx
with i
j < BL2x, (h,k)
Then we have
,. (,u,.)
n&
(h,k)
h(modk)]
J, n
j(x,x;)
and with
/
N
\zj-
0B,N,u
i
(k)
(
z
z-z()(og x)Z- )
x(L2x)j
j!
z=0
(5.3)
(logx)
k, where the coefficients
-N-I
(e
2(3Q(k)))N
z(k)
)
are given by (4.11).
In view of our notation, the orthogonality relations for characters yield
1
w’3(x’k’h’)’
()
xmod7 kX(h) wj(x,x;)
(5.3)
EXTENSIONS OF SOME FORMULAE OF A. SELBERG
i(n)
(2.7)
is completely multiplicative,
modulo k
Since any character
is additive when restricted to squarefree
A (x,x;)
z
and our definition of
297
(n)
is additive, and
(2.3), (2.4),
it follows from
n,
that
Az(X,X;
u.(x,x;)z j
7.
(5.4)
J
j=O
If C
with
is any circle of radius less than B
O, traversed once counterclockwise, then Cauchy’s Residue Theorem implies that
This sum is actually finite.
center
1
j(x’x;)
2-
7C Az (x’x;)z-j-ld z
Now after the proof of Lemma 3, we declared that
of this paper.
So, for nonprincipal
B
would exceed i
for the remainder
C have radius
we can let
X
(5 5)
i and apply
Lemma 6 to obtain
u(X -c3i0g )
x
OB,
Uj(x,x;)
If
x
is principal, then
XO
Hence, we can estimate
we can view
that
ez(Q(k))
N
Tj(X,Xo ;)
x
i
7.
=i
2--
Thus, if we put
fC
z
-i
.6z(k)(lgx)
OB,N, u(x(lgx)-N-I (e2 (3Q))N
+
By (4.12),
by Lemma 7, with
z
-i
k
replaced by
as either the principal character modulo
0
e%z(k)
(5.6)
is also the principal character modulo
0
Az(x,x;)
e
z(k)
is analytic at
Q(k)
Q
z- z-
k
or modulo
In (4.11),
Q(k)
so
we have
+
dz
, x)Re
7clZ-J-ll\--(0)
z= O,
Q(k)
Q(k)
log
(5.7)
z
Idzl
and consequently Cauchy’s Residue Theorem
yields
2i
7C z z(k) (logx)
z-3dz + (j-l)
bz
z(k) (logx)
z
j -i
(5.8)
z=O
If we substitute (5.8) into (5.7), and then substitute (5.6) and (5.7) into (5.3),
we find that it suffices to prove that the remainder term in (5.7) is
Since
j
<
BL2x,
C
we can let the circle
in (5.7) have radius
J/L2x
Thus, the
integral in the error term is
I
Since
Q
k
2
U
.j
3
< exp/log x
exp[ (j
cos8 0
0)
(e2x)
-I
(e
2
Q)
(-log x)]d0
(5.10)
we can conclude from part iii) of Lemma 1 that
0
Hence, since
cos
\0
for
<
0 &
L2(Q(-Q) log x) s L2x
&
/2
and
cosS 0
(5.ii)
for
u/2 & 8
,
we can
298
C.A. SPIRO
bound the integrand by
Thus,
I
/2
for
It is well known that
/2eJ
0
cos
d I0(j)
where
I
cos
ej
,
(e2x)Jj-J(7/2
0
I
0
<
ej
and by
j cos
e
d O(eJj -I/2)
d
cos
0
for
/2.
+ i)
(5.12)
(in fact, we have
is the Oth Bessel function of purely imaginary
0
If we combine this result with (5.12), and apply Stirling’s formula for
argument).
we find that
j!
(L2x)Je j(j!)-I
I
I
Since
(5.13)
(5.7), that error is (5.9), and the
is the integral in the error term in
theorem follows. //
Upon performing the differentiation indicated in our theorem, we find
REMARK.
that we can replace the main term of our estimate for
.
N
(log x)
=i
Where
gj,,k(U)
and
m
i
provided that
requirement that
6.
by
-j ,,k(L2x)
of degree
u
(5.14)
I
j
From Lemma 3, we can deduce that Theorem i is valid for the functions
REMARK.
mu
is a polynomial in
j(x,k,h;)
for any value of
< B< 2
B
B > i,
and that Theorem I holds for
Furthermore, in view of the statement of Theorem i, the
can be replaced by the assumption that
exceed i
B
be positive.
AN ANALOG OF THE SATHE-SELBERG FORMULAE
THEOREM 2.
integers with
Fix
i
B’
0
with
B’L2x,
j
< B’ < B
Select
i, I
(h,k)
k
> 0, and let h,
u
exp logx
<
and
Q(k)
j, k
be
(logx)
u
Then we have
x
(k) logx
.(x,k,h’)
3
uniformly in
REMARK.
F(I
(j-l)!
h, j, and k
where
with
0
B’
0
(j-I
/LzX)
+
OB,B’,u
According to Lemma 3, we can take
< B’
2
if
1
/A]
exp[(ol -)
m
L2x
(6.1)
((L2x)’2 3(16k))2)]
J
B
(L
for any selection of
or
mi
provided that
k.
to be large enough so that our
B’ > 0 if
and for any choice of
fl
+
denotes the principal character modulo
XO
theorem is valid for any choice of
B’
X0;)
2x
(L2x)
Here
B’
satisfying
A
is the constant mentioned
in (1).
PROOF.
We can conclude from (44) and Theorem 1 with
_z;_l
+ 0
(k) (log x)Z-i
[
i
k)
N
1,
that
+
z=O
x(L2x)J
j!
(6.2)
log x)
-2L 2 (3Q(k))
EXTENSIONS OF SOME FORMULAE OF A. SELBERG
299
where
Fz)(--k))Zg(l’z’Xo;)
lz (k)
(6.3)
By the functional equation for the gamma-function, we have
iz (k)
For
I,
j
the theorem follows from (6.2) and (6.4).
(z)
iz(k)
g(l,z,Xo;) is
z
B’ < B,
and since
k,
in
k
and
z
> I,
j
set
Iz[
with
(6.5)
bounded uniformly in
we can deduce from
g’(l,zo,Xo;)
the derivative of an analytic function that
bounded uniformly in
If
k(z,k;)
Since part i) of Lemma 3 implies that
< B, and
(6.4)
g(l’z’Xo;)
F(z+l)
N
B’
z
with
Cauchy’s inequality for
g"(l,z,Xo;)
and
i/F(z + i)
Furthermore,
entire, so that its first two derivatives are uniformly bounded in
z
with
are
is
[z[
N
B’
Hence, it follows from (6.5), (6.4), and part iii) of Lemma i that
[i"(z)[
<B,B
k
uniformly in
(log (3k/(k)))
\
and
z
Izl
with
2
<B,B’
J
(L
3(16k))2
(6.6)
From (6.2), (6.5), and Cauchy’s Residue
B’
Theorem, we can conclude that
n.(x,k h;)
3
1
1
x
(k) logx
2hi
fC (z)(logx)Zdz
+
C denotes the
where
Put
X(r)
-iIf
rz
fC
(log
(j
<
(6.7)
i
x(L2x)J
(k)
j!
l)/L2x
L2(3Q
(K)))
(log x)
2
traversed once counterclockwise.
Then the integral in (6.7) is
l)/L2x
(j
r
[z
contour
OB’u
+
1
x}Zz-Jdz +
7C
[k(z} k(r)
denotes the straight line segment from
k(r)(zr
to
r}]
z,
z
(log x} z -j dz
then we have
__(z-w)h"(w)dw
(z)-7(r)- (z- r)’(r)
(6.8)
(6.9)
rz
for
Iz[
B’
So, it follows from (6.6) and the Triangle Inequality that
lk(z)-(r)-(z-r)’(r)
again uniformly in
and
k
second integral in (6.8).
z
with
<B,B,(L3(16k))2(k(-k--) )
[z
<
B’
Re z
lr-zl 2
(6.10)
We now apply this estimate to the
At the same time, we evaluate the first integral in (6.8)
by Cauchy’s Residue Theorem.
Since the quantity in (6.8) is the integral in (6.7),
we obtain
(e2x)J-i (r)
7c (z) (logx)Zdz
(j-l)
+
OB,B,
+
L3(16K))27
r Iz-Jldz ,)
z
ir- z -----log x
(6.11)
C.A. SPIRO
300
C
By our choice of
we can rewrite the integral in (6.11) as
4r
f0 (i- cos 0)e(J-l)cos 0(k__c.)-r cosd
[0,i/i-] the integrand
most
(sin)/sin(i/I/ ).
On the interval
integrand is at
O(L
integrand is
3-j
2(8k))r)
+
i/I-N/2,
For
the
Finally by part ill) of Lemma i, the
[/2, ].
on the interval
4r3-Jo(eJj -3/2)
O(j-le j)
is
(6.12)
the integral in (6.11) is
Thus
O(exp[(j-l)(L2(8k))/e2x])
(6.13)
We will show that the second error in (6.13) can be absorbed into the first one.
Indeed, if
since
k
(L2x)/L3x
1
j
then
exp[(j-I)L2(8k))/L2 x] e eJ-lj -3/2
Furthermore, if j-1 > (L2x)/L3x, then
x.
k
since
(log x)
u
1
(L2x)
x
(k) logx
j-I
(r) +
(j-l)
By applying Stirling’s formula for
Q(k)
(logx)
u
eJ-lj-3/2
Therefore, if we substitute (6.11) into (6.7)
0B
u-
(L2x) J
x
1
(k)
+OB,B’,u
N
U
(6 15)
In either case, we have shown that the integral in (6.11) is
0B,u(r3-JeJj-3/2)
j (x,k,h;)
u (L2x)B B
exp[BL2(8k)]
exp[(j-l)L2(8k))/L2x
(6.14)
(
L (3Q (k))
2
2
(log x)
j’
x
(k) hogx
r
+
(6.16)
3-jeJj-3/2 (L 3(16k))
to the second error
j!
we find that
term using the fact that
(6.4), (6.5) and the
to estimate the first error term, and recalling
r
definition of
we can now deduce the asserted result.
H
AN APPLICATION TO THE STRENGTHENING OF SOME RESULTS OF G. J. RIEGER
7.
By using Lemmas 6 and 7, we can improve Stze 2 through 7 of Rieger [3] by replacing each estimate by
log x
THEOREM 3.
k
x
times an asymptotic expansion descending by powers of
We illustrate the idea with an example:
(cf. Satz 2 of Rieger [3]): Select
-
< explog x and that Q(k)
(log x)
integer relatively prime to k,
Then we have
E
x
a (n;)
z
nNx
n (mod k)
=-
uniformly in
k,
,
and satisfy (4.12).
N,
u
N
+
and
OB,N,u-r-r’’k@m)
z,
> O, suppose that
and assume that
and let
"i z(k) (log x)
u
N
[z[
B
Let
be any
be a fixed but arbitrary positive integer.
z- +
(7.1)
(logx)
Re z-N-i
where the coefficients
z(k)
are given by (4.11)
EXTENSIONS OF SOME FORMULAE OF A. SELBERG
PROOF.
301
According to the orthogonality relations for characters, we have
.
i
(k) xmod k X(2’)
(n;)
a
n-x
A (x,x;)
z
(7 2)
2’ (rood k)
n
//
f(s,z,x,) by (L(s,x)) z we can argue as in Lemma 5
d (n)x(n) of the same form as the expression on the
Z
This theorem is now an immediate consequence of Lemmas 6 and 7.
Furthermore, if we replace
to obtain an estimate for
YriSE
right side of (4.10).
Then, we can reason as in the proof of Theorem 3, but using
Hilfsatz 5 of Rieger [3] in place of Lemma 6, to arrive at the following result:
(cf. Satz i of Rieger [3]):
THEOREM 4.
Under the hypotheses of Theorem 3, we
have
.
nx
n
=
N
X
d (n)
z
’(k)
2’ i
2’ (mod k)
2’z(k)
(log x)
k, 2’, N,
Y2,z (k)
+
(7.3)
+
uniformly in
.--2’
and
OB,N,u
x
(log x)
Re
z-N-I(L2(3k))N(
z, where the coefficients
(2,-1)!F(z-2’+l)
k-0--l
2’z(k)
]
are given by
(7.4)
s=l
Furthermore,
Iz2’z (k) <,B (L2(3k))2’’
uniformly in
k
and
(7.5)
z
ACKNOWLEDGEMENT.
The author would like to thank G. J. Rieger for enquiring about what
improvements could be made in Rieger [3] by using the results of this paper Finally,
we gratefully acknowledge the help of a referee, who made a number of helpful comments
in regard to our work.
REFERENCES
I.
SELBERG, A. Note on a paper by L. G. Sathe, J. Indian Math. Soc. (N.S.) 18 (1954),
83- 87.
2.
DELANGE, H.L. Sur des Formules de Atle Selberg, Acta Arith. 19 (1971), 105-146
(errata insert).
3.
RIEGER, G.J.
4.
RIEGER, G.J.
5.
NORTON, K.K. On the Number of Restricted Prime Factors of an Integer, II.
Acta Math. 143 (1979), 9-38.
6.
SPIRO, C.A.
7.
8.
Zum Teilerproblem
yon
Atle Selberg, Math. Nachr. 30 (1965), 181-192.
Uber die Anzahl der als Summe yon zwei Quadraten darstellbaren und
in einer primen Restklasse gelegenen Zahlen unterhalb einer positiven
Schranke. II, J. reine angew. Math. 217 (1965), 200-216.
How often does the Number of Divisors of an Integer Divide its
Successor?, J. London Math. Soc. (to appear).
TITCHMARSH, E. The Theory of the Riemann Zeta-Function, Oxford, at the Clarendon
Press, 1951. 346 pp.
WALFISZ, A. Zur additiven Zahlentheorie. II, Math Zeit. 43 (1936), 592-607.
C.A. SPIRO
302
9.
Introduction to Modern Prime Number Theory, Cambridge Tracts in
Mathematics and Mathematical Physics, no. 41, Cambridge, at the University
ESTERMAN, T.
Press, 1952.
I0.
x
+ 75
pp.
WHITTAKER, E.T. and WATSON, G.N. A Course of Modern Analysis, Cambridge University
Press, fourth ed. reprinted, 1963; original fourth ed., 1927, 608 pp.