Proc. Indian Acad. Sci. (Math. SO.), Vol. 103, No. 1, April 1993, pp. 1-25.
9 Printed in India.
Transformation formula for exponential sums involving Fourier
coefficients of modular forms
C S YOGANANDA
Department of Mathematics, Indian Institute of Science, Bangalore 560 012, India
MS received 10 June 1992; revised 14 November 1992
Abstract. In 1984 Jutila [5] obtained a transformation formula for certain exponential sums
involving the Fourier coefficients of a holomorphic cusp form for the full modular group
SL(2, Z). With the help of the transformation formula he obtained good estimates for the
distance between consecutive zeros on the critical line of the Dirichlet series associated with
the cusp form and for the order of the Dirichlet series on the critical line, [7]. In this paper
we follow Jutila to obtain a transformation formula for exponential sums involving the
Fourier coefficients of either hoiomorphic cusp forms or certain Maass forms for congruence
subgroups of SL(2, Z) and prove similar estimates for the corresponding Dirichlet series.
Keywords. Exponential sums; summation formula; cusp forms and Maass forms;
transformation formula.
1. Introduction
By an exponential sum we mean a sum of the form
a(n)e(f(n)),
M<n<M"
where a(n) is an arithmetical function and f is a real valued function on [M, M'].
Many a problem in number theory reduces in the ultimate analysis to estimating an
exponential sum. The Waring's problem, the Goldbach's conjecture, the Dirichlet
divisor problem and the order of the Dirichlet series in the critical strip are very
good examples of this phenomenon.
The most commonly employed method to estimate such sums is due to Van der
Corput and Vornoi. The basic idea here is to transform an exponential sum into a
new shape by first converting the sum into an integral--Van der Corput's lemma
and Vornoi summation formula--and then evaluating the integral by the 'Saddlepoint method'. Since then various summation formulae of the Voronoi type have
been found; a very good survey is to be found in [2]. In 1984, Jutila [5] discovered
that by replacing f(n) by f(n) + rn where r is an integer (which does not affect the
sum) before transforming the sum one was led to much better transformed sums.
Another important observation made by Jutila was the flexibility of this method
which, he showed, works in the case when a(n)'s are Fourier coefficients of a cusp
form of weight k for the full modular group, SL(2,Z). With the help of this
transformation formula he was able to obtain for the Dirichlet series associated to
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C S Yogananda
cusp forms for SL(2, 7/) analogues of many results known in the case of the Riemann
zeta function, ((s), like the distance between consecutive zeros on the critical line, the
order on the critical line, mean square estimates and higher power moments I-5,6, 7].
In this paper we show that the transformation formula and the above mentioned
applications carry over to the case when a(n)'s are Fourier coefficients of either
holomorphic cusp forms or certain Maass forms (Maass forms f with Af = 1/4f) of
higher levels. While some of the above mentioned applications were already known
in the case of cusp forms for SL(2, Z) due to Good [4] they seem to be new in the
case of cusp forms of higher levels.
Mention must also be made of the work of Meurman [8, 9] who has extended
some of the results of Jutila to the case of L-functions associated to Maass wave
forms for SL(2, 7/). Presumably his work also extends to higher level Maass forms.
The class of Maass forms we consider in this thesis does not occur at level one.
2. Functional equations and summation formulae
In this section we are concerned with functional equations for Dirichlet series (and
their twists by additive characters) associated to certain modular forms for congruence
subgroups of the full modular group SL(2, 7/). The class of modular forms we consider
consists of all holomorphic forms and the subclass of Maass forms with 1\4 as the
eigenvalue of the Laplacian on the upper half plane. We consider the holomorphic
case (Theorem 2.1) and the case of Maass forms (Theorem 2.2) separately. The
summation formulae these functional equations lead to are written down at the end
of the section.
If f is a function on the upper half-plane H, k an integer and A is in GL § (2, R)
then final(T) will denote the following function
(detA)k/Z(cz+d)-kf(az+~b']\cr+a/ where A = ( ~ ) .
Holomorphic case
For k, N i> 1 integers and e a character mod N let M(N, k, ~) denote the space of
modular forms of level N, weight k and character e. Thus iffeM(N, k, e) and AeFo(N )
we have
Note that M(N,k,e) = {0} unless ~ ( - 1) = ( - 1)k, and that M(N,k, 1) is M(N,k), the
space of modular forms of weight k for Fo(N ). If H(N) denotes the matrix (o -~) then
k
f fltmm~
defines an isomorphism from M(N,k,e) onto M(N,k,~ where ~ is the
(complex) conjugate character.
Let feM(N,k,e) and f(z)= ~ a(n) exp(2ninz) be its Fourier expansion at the
n=O
cusp i ~ . We are interested in functional equations for the Dirichlet series
q~y(s, h/m) =
oo a(n) exp(2zrinh/m)
n= I
n s
Transformation formula for exponential sums
3
Theorem 2.1. (a) Case when (m, N) = N. Let
dO(s,h/m) = (m/2rr) s F (s) ~bf(s, h/m).
Then
a(O) ike(h)a(O)
+ - - ,
s
k-s
is EB V (entire and bounded in every vertical strip) and we have the functional equation:
dO(s,h/m) +
dO(s,h/m)= ike(h)dO(k- s , - h/m), where h is defined by h h - 1 (modm).
k
(b) Case when ( m , N ) = l . Let f[[mml(z)=g('c)~M(N,k,e-)
and g ( z ) =
~ b(n)exp(2~inz) be it's Fourier expansion. Further let
n=O
dO(s, h/m) = (mx/N /2~)SF (s) qSy(S, h/m) and
tF (s, h/m) = (mx/ N /2;t)* F (s) (a9(s, h/m).
Then
do(s,h/m)+(m2N)_S/2(a(Os) d e(m)b(O)~k_s J
is EBV and we have the functional equation:
dO(s,h/m) = e(m) ~ (k - s, - Nh/m).
[]
Proof. (a) Let t > 0 be a real n u m b e r and put z = h/m + i/mt and z' = - him + it~re.
Then z and r' are in the upper half-plane and are equivalent under Fo(N) by the
matrix (remember m - 0 (mod N))
A = [ hm (hhh-1)/m]
i.e. A (z') = z. Therefore we have f(z) = e(h)(it)kf(z '). If Re(s)is sufficiently large we have
dO(s,h/m) =
a(n)exp(2ginh/m)
p - t e x p ( - 2~nt/m)dt.
n=l
=
p-1 {f(h/m + it~m) - a(O)} dt.
=
: - 1{f(h/m + it~m) - a(O)} dt -
a(O): -1
!
+
;o
P- Xf(h/m + it/m)dt.
N o w consider
f
/ t 5- If(him + it/m)dt = f f t-s- if(hi m + i/mt)dt.
= e(h)i k
tk-~- l f ( z ' ) d t .
1
4
C S Yogananda
Thus we have
r
h/m) = f { I f ( h i m + it/m) -
a(0)] t ~ -
1
+ e(h)ik[f( - h/ra + it/m) - a(O)]t k-s-1 }dt
a(O)
e(g)ika(O)
S
k--s
This integral representation proves the claims made in (a).
(b). For x~R let at(x) denote the matrix (o1 x1). Let t > 0 be a real number and set
z = h/m + i/m2Nt and z ' = - f i l m + it. We need to know f(z) in terms of g(z'). For
that first observe that
~(h/m)H(m 2 N) = H(N)
(o
o)
_ Nh
m
where b is defined via N h b - = - 1 (mod m) (which is possible because (re, N ) =
m -Nh n b ) is in F~
(m'h)=l')andnisch~
f(T)
= ( m 2 N) k/2 (it)kfl[~(h/m)]
= (m 2
Theref~
we have
[H(Nm2)](it).
N) k/2 (it)kfl[ntN)] [(_%?)] [~(b/m)](it).
= ( m 2 N) k/2 (it) k g(n)g I[ot(b/m)](it).
= (m2N)k/Z(it)ke(m)g(-- Nh/m + it).
Note that we have made use of
(i)
g(n) = e(m) as mn = 1 (mod N),
and
(ii) exp(2nib/m) = e x p ( - 2niNh/m)
since
Nhb = - 1 (mod m).
Consider now the following integral representation:
@(s, h/m) = (m 2 N) ~/2
[f(h/m + it) - a(O)] P- 1dt.
rr"
=(m2N) ~/2
L
+ Jo
[f(h/m+it)-a(O)]t~-ldt -
a(0)t~-ldt
dllm~/N
f ( h / m + it)t s- 1dt
.
Now
f o1/mx/N
oo
f (h/m + it) P - 1dt = (m z N)-~ |
= (m 2 N) (k/2)-~e(m
f (h/m + i/m 2 Nt) t - ~- i dt,
g ( - Nh/m + it)tk-~- 1dt.
Transformation formula for exponential sums
5
Thus we get
9 (s, h/m) = (m 2 N) ~/2
{ [f(h/m + it) - a(O)] : - 1
J1/m~/N
+ e(m)( m2 N) ~k/2~-~[0(-- Nh/m + i t ) - b(O)] t ~-~
- ~ } dt
_(m2N)_./2(a(Os ) + e(m)b(O)~
This integral representation verifies the claims made in (b). Thus the proof of the
Theorem is complete.
Remark. As these functional equations characterise modular forms of level N (see
[12] and [13]) we cannot in general hope to get similar functional equations when
1 < (m, N) < N. For instance if f is a new form of level N then existence of functional
equations for 1 < (m, N) < N would mean that f is a form of lower level which it is not.
Non-holomorphic case (see I-11])
Let A=--y2(f~2/t~X2 +~2/t~y2) denote the Laplacian on the upper half-plane H
associated with the hyperbolic metric. Let N/> 1 be an integer, e a character mod N
and )~ a complex number. Let f be an even Maass form of level N, character e and
eigenvalue (for A) 2. This means:
(i) f e L2(F1 (N)\H);
(ii) f(?x) = e(d)f(z) for ~ -- (ca b~)el'o(N),
(iii) A f = 2f, ). = 1/4 + r2;
(iv) f is a simultaneous eigenfunction of the Hecke operators Tn,
(n, N) = 1 and T_ if(z) = f ( -
z-) = f(z).
Such an f has a Fourier expansion of the following type:
f(~) = ~ a(n)yl/2Ki,(27tny)cos2~nx,
z = x + iy.
Let f be an even Maass form with 2 = 1/4 i.e. r = 0 with Fourier coefficients a(n).
Here again we are interested in functional equations for the following Dirichlet series
associated with f :
~f (s, h/m) =
,
~ f (s, h/m) =
n=l
a(n)cos2~nh/m
n"
~ a(n)sin2~tnh/m
ns
n=l
and in this direction we have:
Theorem 2.2. (a) (m, N) = N. The functional equations are
~l~y(s,h/m) = e(h)~:(1 - s, - h/m)
r
h/m) = - e(h)~f(1 - s, - h/m),
6
C S Yogananda
where hh=-- 1 (rood m)
9 I(s, h/m) = (m/n)~F 2 (s/2) ~by(s,h/m),
and
O'r(s, h/m) = (m/~)sF 2 ( ~-~-~ ) tk'/(s, h/m)
(b) (re, N)= 1. Let fl~m~j(z ) = g(z); then g is an even Maass form of level N and
character ~. Let b(n) be the Fourier coefficients of g, Then the functional equations are:
Oi(s, h/m) = e(m)Og(1 - s, - Nh/m),
%(s, h/m) =
- s, - N h / m )
where
9 z (s, him)
= (Trlmx/N)-" F 2 (s/2) dps(s, h/m)
and
O~(s, h/m) = (rc/m~fN)-SF 2 ( ~--~- ) r
h/m).
[]
Proof Let fx(z) = 1/2nid/dxf(z); (~ = x + iy). Then we have
f x(z) = i ~_a(n)nx/ y Ko(2nny)sin(2rmx ).
We need to know how f~(z) transforms under the transformations A and H(N). For
this let fy = 1/27tia/Oyf(z). If U e F o ( N ) then f(Uz) = e(d)f(z), U = ( ad cb ) . Soweget
f~(z) = 1~2hi d/Ox f (z) = e(d)/2ni d/~x f (Ut)
= e(d)[f~(U~)Re(aUz/ax) +fy(U~)Im(dU~/dx)]
= e(d)[fx(Uz)Re((cz + d)-2) + fy(Uz)Im((cz + d)-2]
Now taking z = h / m + i / m t and A as in the above proof we see that
fx(h/m + i/mt) = e(h)(- t)- 2f ~(_ him + it~m). We get a similar transformation formula
under H(N).
The rest of the proof is along the same lines as that of theorem 2.1 (and so we will
not reproduce it here) but will use the following formula to get an integral
representation for @(s, h/m):
1/4(m/gn)~F2(s/2) =
;o oKo(2~ny/m)y,- i dy.
Summation formulae
We begin by stating a theorem of Berndt [3]. Let {2,} and {~,}-be two sequences
of positive numbers strictly increasing to infinity. Let {a(n)} and {b(n)} be two
sequences of complex numbers, not identically zero such that the Dirichlet series:
d~(s) = ~a(n)~.-] ~ and 0(s) = ~b(n)~2 ~
converge in some half-plane. Suppose further that they satisfy the functional equation:
Transformation formula for exponential sums
7
Z(s) ~(s) = z(r - s)~b(r- s) where •(s) is one of the following three gamma factors:
(i) F(s) and r arbitrary real;
(ii) F(s/2)F(s - p/2) where p is an integer and r = p + 1;
(iii) F2(s + 1/2) and r = 1,
Also further, suppose that the poles of Z(s)(~(s) are confined to some compact set.
Define for x > 0, Qq(x) and I~(x) as follows:
1 fc F(s)q~(s) xS+qds
Qq(x) = ~ __q F(s + q + 1)
where Cq is a cycle enclosing all of the integrand's poles; and respectively as X(s) is
as in (i), (ii) and (iii):
l~(x) = x~" + ~)/2.I + ~(2x/x)
= x~p+~+ 1)/2{cos(rt(p + 1)/2)dp+q+ 1(4x/x)
-sin(rt(P+ l)/2)[Y'+~+'(4x/x)+/2(-_l)~
\ 7t J K "+~+~(4x/x) }
= x'.q+"/2 { Yq+l (4x/x) + ( 2(-!)q+ l ) Kq+ l (4x/x) }.
Theorem 2.3. (Berndt [3]): Let fECtl)(0, oo). Then
~'
a(n)f(;.,)=
Q;(Of(t)dt + ~
,_-~
a<a~<b
[]
I ~(~t)f(t)dt
~n
Summation formulae for the holomorphic case
For the sake of simplicity we write down the summation formulae only when f is a
cusp form. In this ease only the second term on the right hand side of the general
summation formula in theorem 2.3 will survive for then cb(s) is entire. Accordingly
let f be a cusp form of level N and character e. The functional equations of theorem
2.1 give rise to the following summation formulae:
(a). Case when (m, N) = N.
a(n)exp(Eninh/m) f (n) = ike(h) 2It ~'~a(n)exp(-- 2~rinh/m)n-tk- 1)[2
a<~n<~ b
X
m
(b). Case when (m, N)= 1.
a(n)exp(2rtinh/m) f (n) = g ( m )
~ b ( n ) e x p ( - 2rciNh/mln -~k- 1)/2
a<~ n<~ b
b
[4~(nx)I12"~
Io x~'-"/"l'-'~, ~
. . . .
):,x,,x.
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C S Yogananda
Summation formulae for the non-holomorphiccase
Let f be an even Maass form with 2 = 1/4 with Fourier coefficients a(n). Then the
functional equations of theorem 2.2 imply the following summation formulae (note
that p = 0):
(a). (m, N) = N.
a(n)cos(2nnh/m)f(n) = n-~)n
st
~ a ( n ) c o s ( - 2nnh/m) x
a~n~b
/m
Ij~ I 2/nKo( 4n~-~(mnX)) - yo( 4n~--(mnX)) ]f (x)dx,
~, a(n)sin(2nnh/m)f (n) = 8(h~)~Ea(n)sin(-2rm~/m) x
a <<.n <~ b
zm
obF Yo( 4n~((nx,~ + 2/nK ~( 4n~/(nx, ~ l,
L \
m /
\
m /3
(b). (m, N) --- 1.
e(m!n
~, a(n)cos(2nnh/m)f (n) = 2m~/
N ~ b ( n ) c o s ( - 2nnR~/m) x
a<~ n<~ b
) -
ro(4
'2/lf(x,
dx.
,mv,,/_
1
~, a(n)sin(2nnh/m)f(n)= e(m!n Eb(n)sin(_ 2zRg/m)x
, <~.~ ~
2mx/N
I~[ Y~
+ 2/nK~
3. Transformation formulae
In this section we obtain, following Jutila [6], a transformation formula for
exponential sums of the type
~, a(n)o(n)exp(2nif(n))
a<~ n<<.b
where a(n)'s are Fourier coeff•
of either a holomorphic cusp form or a Maass
form with 2 = 1/4 for a congruence subgroup of SL(2, Z) and f and g are functions
on [a, b]. To begin with we recall results on exponential integrals due to Atkinson
and Jutila. This is essentially chapter 2 of [6] without proofs. Then we obtain
transformation formulae for exponential sums involving Fourier coefficients of cusp
forms considered in w1. We conclude this section with some special cases of the
transformation formula for Dirichlet polynomials associated with these cusp forms.
Transformation formula for exponential sums
9
Exponential integrals
Let
g ( x ) e x p ( 2 n i ( f ( x ) + ctx))dx =
I =
h(x)dx.
For a positive integer J and a positive real number U define a smoothed version Ij
of I by:
U-
1
du I
dO
dua
h(x)dx =
Fla(x)h(x)dx,
u = u 1 + ... + ua.
~a+u
Also let Io = I. Note that 0 < r/(x) ~< 1 for x~(a, b) and t/(x) = 1, for a + J U <~x <~b - JU.
We quote three theorems below first of which gives an approximate value of the
integral I in terms of saddle points (Atkinson), the second theorem its generalization
to Ij due to Jutila and the third gives an estimate of I~ when f has no saddle points
in (a, b). For proofs of these theorems see 1,6].
Let f and g be functions on I-a,b] satisfying the following conditions:
(i) f is real for a ~<x ~<b;
(ii) f and g are holomorphic in the domain
D = {zllz- xl < # for some xe(a, b)} where/l is a positive real number;
(iii) there are positive numbers F and G such that:
Ig(z)l << G and If'(z)l <<F# -x for zeD;
(iv) f"(x) > 0 and f"(x) >>F# -2.
Since i f ( x ) > O, f ' ( x ) + ~ is monotonically increasing and hence has at most one zero
in (a, b), say x0. Further let
Ej(x) = G ( l f ' ( x ) + atl + f " ( x ) l / 2 ) - ~ - 1
Theorem 3.1. Let f and g be as above. Then
I = g ( x o ) f " ( X o ) - 1/2 e x p [ ( 2 n i ( f ( X o ) + otxo + 1/8)]
+ O(Gexp [ - A Ictl# - AF](b -- a)) + O(G#F-3/2) + O(Eo(a)) + O(Eo(b))"
[]
Theorem 3.2. Let U > 0, J >10 be a f i x e d integer, J U < ( b - a)/2 and f and g be as
above with the additional condition that F >> 1. Suppose, also that U >>/~F- 1/2. Then
with I z as above we have:
Ia = r
1/2 exp(f(xo) + ~tXo+ 1/8)
+ O((1 + (#/U)J)Gexp( - al~l# - a F ) ( b - a))
-I- 0((1 + F1/2)G#F -3/2)
10
C S Yogananda
where ~(Xo) = 1 for a + J U < x o < b - JU,
~ ( X o ) = ( J ! U S ) -1 ~
( - 1)j
~
j=0
cvf"(Xo)-V(xo-a-jU)
s-2v
0~<v~<J/2
for a < x o <<.a + J U with j l the largest integer such that a + j l U < Xo
~ ( x o ) = ( J ! U S ) -1 ~
( - 1)~
j=O
~
c v f " ( X o ) - V ( b - x o - j U ) s-2~
0<~ v<~J/2
for b - J U <. x 0 < b with J2 the largest integer such that b - J 2 U > x o. The c~ are
numerical constants.
[]
Theorem 3.3. Suppose f and g are functions satisfying (i) and (ii) above. Assume further
that I9(z)l<<G, If'(x)[M, and If'(z)l<<M for zeD and x~(a,b). Let I s denote the
smoothed version of I with ~t = 0 and 0 < J U < (b - a)/2. Then
I s << U - 1 G M - S - x + (#s U 1 - s + (b - a)) G e x p ( - AMp).
[]
Transformation formulae
Before we proceed to the theorem we quote a Lemma (without proof) which
summarizes the properties of Hankel functions we need to use.
Lemma 3.4 Let 61 < ~ and 62 be fixed positive numbers.
[argz[ ~ n - 61, Izl/> ~2 we have
Then in the sector
H~J~(z) = (2/nz) 1/2 e x p ( ( - 1)-/- i i(z - nn/2 - re/4))(1 + g~(z)),
where the functions 9j(z) are holomorphic in the slit complex plane z r O, [ar9 z[ < rt,
and satisfy 1Oi(z)[<< Iz[- 1 in the above sector. Further we have
J,(z) = 1/2(H~,l)(z) + H~2J(z))
and
Y,(z) = 1/2i(H~,l)(z)- H~2)(z)).
W e also have
K , ( x ) = (n/2x) 1/2 e x p ( - x)(1 + O ( x - 1)).
[]
In what follows, 6 denotes an arbitrary small positive constant not necessarily the
same in each occurrence. Put L = log M.
We have the following theorem which gives a transformation formula in the case
of holomorphic cusp forms. Accordingly let
f(z) = ~
a(n)exp(2rcinz)
n=l
k
be a cusp form of level N, weight k and character e. Further let fltn(ml(z)
= g(z) =
Eb(n) exp(2ninz).
Transformation formula for exponential sums
11
Theorem 3.5. Let 2 ~<M 1 < M 2 ~<2M 1 and let f and g be holomorphic functions in
the domain
D=
{zllz-xl
<cM1 for some xe[M1,M2]},
where c is a positive constant. Suppose that f ( x ) is real for x in [M 1, M2]. Suppose
also that, for some positive numbers F and G
Ig(z)l << G,
If'(z)l << F M - 1 ,
for zeD, and that (0 <)f"(x)>> FM-~ 2 for x e [ M 1, M2].
Let r = h/m be a rational number such that
1 <~m<<M~/2-a,
i r t ~ F M ~ 1,
and
f'(M(r)) = r
for a certain number M(r) in [M 1, M2]. Write M s = M(r) + ( - 1)Jmi, j = 1, 2.
Suppose that m l ~ m 2, and that
M~ max(M 1F - 1/2, Ihml) <<ml << M~ -a
Define for j = 1, 2
f(x) -- rx + (-
Pi,.(x) =
I)i-1/'2x/(nx)
~k
f ( x ) - r x + ( - 1 ) 1 - 1 \ mx/N
(k
1)
4
' if(re, N ) = 1.
and
~(r--f'(Mj))2m2Mj, if(m,N) = N;
ni = ( ( r - f ' ( M i ) ) 2 ( m x / N ) ) 2 M j if(re, N) = 1.
and for n < n let xj,. be the (unique) zero of p),n(x) in the interval (M1, M2). Set
~ ike(h)(2m) -1/2, if(re, N) = N and
A = (e(m)(2m~/N)_l/2 ' if(re, N)-- 1.
Then we have
a(n)g(n)e(f(n)) =
MI <~n<~M2
2
(k/2)-(3/4)
A ~ (-- 1)j - I Z a'(n)e(nh'/m) n-tk/2)+ 1/4 xj,n
j=l
g(Xj,n) X
n<ni
py,n(Xj,n)- l/2 e(pj,n(Xj,n) -t- 1/8) + O(G(Ihlm)Xl2 M~lk-1)/21"1r
L2) -k-
O(F 1/2 Glhl- 3/4m5/4 Mflk- 1)/2m11/4 L).
where a'(n) = a(n), h' = - h if (m, N) = N and a(n) = b(n), h' = - / V h if (m, N) = 1. []
12
C S Yogananda
Proof. Without loss of generality suppose that r ( = h/m) is positive. Assume that
(m, N) = N; the case (m, N) = 1 is entirely similar. The transformation formula should
be understood as an asymptotic result wherein M t and M2 are large. Before we start
on the proof, we shall note various estimates that are needed; like, for instance, the
order of nj. First note that f " ( x ) ~ F M -2 (for, by assumption, f"(x)>>FM -2 and
the reverse inequality follows from the estimate for f ' and holomorphy of f), and
F >>MI1/2+~ (for F >> M 1r t> m- 1M1 >>M~/2 +~). Thus we have
tr - f ' ( M i ) J ~ m j F M 1 2 (for f'(M(r)) = r).
This gives us the estimate: n ~ F 2 m 2 M 1 3 m 2.
The nfs are determined by the condition p~,.(M~)=0. This implies that for
n<n~,p'j..(x) has an unique zero in (M1,M2). For clearly (-1)~p~. ( M ~ ) > 0 and
( - 1)Jp),.(M(r)) < 0, if n < nj. Note also that Xl, ~ < x2.. and that p),.(x) has no zero
in (M1, M2) if n > n~. Uniqueness of xj,. follows from that p~,.(x) has the same order
as f"(x) if MI is sufficiently large and hence is positive for f"(x) is positive.
Let
S = S(MI, M 2) =
~
a(n)g(n)e(f(n)).
Mt~<n<<.M2
We first replace S by its smoothed version S':
S' = U- 1
where
fo
S(u) =
S(u)du,
~
a(n)g(n)e(f(n))
M l +u<~ n<~ M 2 - u
and U is a parameter to be chosen later. For now we only assume
M~ << U <<.1/2min(ml,m2).
The estimate a(n) << n{k- ij/2 +~ implies that S - S' << GUM~k- 1j/2 L. The choice of the
parameter U later will show that this error has been accounted for in the statement
of the transformation formula.
The idea is to apply the summation formula to S(u) and evaluate S' by using
saddle-point theorems. But instead of applying the summation formula to S(u) as it
stands it has been observed by Jutila that we get better results if we introduce an
exponential factor without disturbing the sum. Accordingly before applying the
summation formula we modify the sum S(u) as:
S(u)=
~
a(n)e(nr)g(n)e(f(n)-nr),
a<~ n<~ b
Applying the summation formula of w2 we get:
S(u) = i~e(h) 2~ ~ a(n)e(- nh/m)n -{k- i)/2
m
a=M l+u,
b=M 2-u.
Transformation formula for exponential sums
13
Now write Jk- 1( ) in terms of the Hankel functions to get:
S(u) = ike(~ ~ a(n)e(- nh/m)n-(k- 1)/21n,
where
ln = ~ f ~ x(k- l)/2[ H~l)_l ( 4rcx/~(mnX)) + H~2)_l ( 4r~x/--(mnX)
) ]g(x)e(f (x) - rx)dx
whence by lemma 3.4 we get I n = 1(1)n+ i~2)
with
l(J) = (2m,/n)- l/2 f f xlk/2)-(a/4)g(x)I I + gj( 4nx/--(mnX)) ]e(pj,n(x))dx.
It can checked that the conditions of the theorems 3.1 and 3.3 are satisfied with - r
in place of r and f(x) replaced by:
f(x) + (-
1)j -
l(2x/(nx)/m - (k -
1)/4 - 1/8),
and/z = M1.
The number x j,n is by definition the saddle point for 1(7) and it lies in the interval
[M1,M2] if and only if n < nj. However, in I~j) the interval of integration is
[a,b] = [M1 + u, M 2 - u ] , and xj,n~[a,b] if and only if n < nj(u) where
nj(u) = (r --f'(M i + (-
1)j - I u)2m2(Mj §
(__ 1)j- I u).
But for simplicity we count the saddle point terms for all n < nj for this frees the
saddle point terms from depending on u and thus we will have the same saddle point
terms for all S(u) and hence for S' as well. The number of extra terms counted will be
<< 1 + n j - n~(u) << 1 +
F2m2M-13ml U.
The saddle point term for I.~ for n < nj is:
(2m)-l/2n-1/4 xj,n
(k/2)-(3/4) X
" X
-1/2
g( j,n)Pj,n(
j,,,)
e(pj,,,(xj,,,)
+ I/8)(I+ gj(2(nXj,n)l/2/m).
Thus up to g~( ) we have the explicitterms claimed in the theorem. The effectof the
omission of gj( ) is:
<<
F- 1/2Gml/2 M~/2)-./4)~, a(n)n-(~/2)-"/4)
n<nj
<<F'-l/2(~rml/2~f(k/2)-(1/4)nl/4
.
.
.
.
.
.
.
.
1
"'j
<<
Gin11/2 M1(k- 1)/2 L.
This error can be absorbed into the first O-term in the formula given in the theorem.
The extra saddle-points counted while replacing nj(u) by nj contribute
<<(1 § F 2 m 2 M 1 3 m x
<< F 1/2 Gh- 3/2 m1/2 m
U ) F - 1/2 G i n - 1/2 M(ik/2)+(1/4)+~n 1 1/4
11/2 M(1k - 1)/2 +e § F - 1/2 Gh3/2 m - 1/2 m11/2 M{1k - 1)/2 +~ U.
Here the first term is absorbed into the second O-term in the transformation formula
and later U will be chosen so that the second term above also goes into the second
O-term.
14
C S Yogananda
We shall now consider the error terms of theorem 3.1 which was applied to I t/)
n
for n < nj. The first error term is clearly negligible. The contribution of the second
O-term is:
<<F- 3/2Gm-1/2 M(xk/~)+(1/4) ~., a(n)n-lk- l)/2n -1/4
/1<</11
<<Gmm3/2 M(lk-1)/2-(3/2) L<< Gm 11/2M l(k-1)/2 L
which again goes into the first O-term of the theorem. The terms O (E 0 (a)) and 0 (E 0 (b))
are similar and so it is enough to consider one of them, say O(E o(a)). This error term is
<<Gm-
r~ a ) 1 / 2 ) - 1 .
1/2Mtxk/2)-(a/4)n- l/4(ip,,,,(a)l + p~,,,(
Consider the c a s e j -- 1; the casej = 2 is even simpler for
" "a") = F - i r 2. Therefore we have
0 and PL,[
p2.,(b) cannot
be very small.
p'x,,~u)(a)=
' a
(IPl,.()1 +
~ F1/2r-xf~
P'~,,(a)l/2)- ~<< [ mM~/2 n[/2 In -
n(u)l~-1 otherwise.
Thus we get that the contribution to S(u) of these error terms is << G(hm)1/2m11/z
M~k- 1)/2 L 2, which goes into the first error term of the formula.
We are now left with showing that the tail part in the summation formula, that is
terms for n > nj, are accounted for in the theorem. Here we make use of theorem 3.2
for the estimation of the exponential integral since for n > nj the integral has no
saddle-points in the interval of integration. Here U will be the smoothing parameter
with J = 1. The contribution of I~~ to S' equals
2
a(n)n-(k-1)/2e(--nh/m) n-1/4 x
(2m) -1/2 E (-- 1)j-1 ~
j= 1
n<nj
f f ~l (X)x(k/2)-(3/4)g(x)[ l § gj(4~(mnX) ) le(pj.n(x,)dx
where ~/l(x) is the weight function. Apply theorem 3.2 with pj.,,(z) in place of f(z)
and /t~.m 1. Note that the conditions of the theorem 3.3 are met if we choose
M = m-1M-~l/2n1/2. The second term on the right hand side of the estimate given
in theorem 3.3 is exponentially smalll and hence can be neglected because
M/z >>m - 1 M~- 1/2 nl/Zmx >>(n/n 1)1/2Fm 2M~-2 >>(n/n 1)1/2Max.
The term corresponding to
U-1 GM-2
<< Om3/2M~~/2~+"/4~ U-
therein is
1 ~ a(n)n-~-./2-~5/4~
n))n 1
<<Gma/2M~k/2)+(x/4)U-1nl- 1/4 L
<<G F - ~/2 toMSk/2)§ ~m ~ 1/2 U - 1 L
<< GFh-
3/2mSI2Mtxk-1)/2m~ 1/2 U -
1 L.
Transformation formula for exponential sums
15
Thus proof of the theorem is complete up to the following error terms:
GUM~k-1)/2L+F-1/2Gh3/2m-I/2 m11/2M (k- 1~/2+~ U
+ GFh- 3/2 mS/2 M~k- 1)/2m ~ 1/2 U- 1 L
The first and the last terms above coincide with the last term in the transformation
formula if we choose U = F1/2h-a/4mS/4m~ 1/4. Then the second term above is
<< Gh3/4m3/4mll/4M 1 << G(hm) 1/2 m 1/2
1 M 61
which can be seen to go into the first O-term of the transformation formula. It only
remains to be shown that the above choice of U satisfies our requirement:
M~ << U <~ I/2min(ml,m2). We have
Um;
1 <<
U(Mll+ a F - 1 / 2 ) - 1 <( (hm)l/am;
1/2Mla << M-~a.
For the other inequality
U>>F1/2h-a/4mS/4M-~(1/4)+6 >>Ml1/4+~ h -1/4 m 3/4 >>M1.a
This completes the proof of theorem.
In the case when f(z) = Za(n)x/yKo(2nny)cos(2nnx) is an even Maass form of level
N and character e as in w2, we have the following transformation formula.
Theorem 3.6. Under the notations and assumptions of theorem 3.5 with k = 1 we have
a(n)g(n)e(f(n)) = A ~ ( - 1)J- 1 ~ b(n)e(nh'/m) x
M l <~n<~ M 2
j= 1
n<nj
n- 1/4xj-.1/'*g(x~,.)p~,.(x~,.)- 1/2e(pj,Jxi, .) + 1/8) +
O(G(Ihlm)1/2m11/2 L 2 ) +
O(F 1/2Glhl- 3/4mS/4 M11/l~ 11/4 L).
[]
Proof. Note that the second error term above is slightly worse than the corresponding
error term in theorem 3.5. This is because the Deligne's estimate which was used in
theorem 3.5 has not been proved for non-holomorphic forms and the best known
estimate is a(n)<< n 1/5 +~. Let S, S' and S(u) be as in the proof of theorem 3.5; further
assume that (m, N) = N, the other case is similar. Thus
S(u)=
~, a(n)g(n)e(f(n))
a<~ n<~ b
=
~
a(n)[cos(21mr)+ isin(2nnr)]o(n)e(f(n)-- nr)
a<~ n<<.b
= St(u) + iS2(u), say.
We now apply summation formulae of w2 to S 1(u) and S2(u) and proceed to evaluate
16
C S Yogananda
the integrals as before.
SI(u)= ~ a(n)cos(2rmr)g(n)e(f(n)-nr)
a<<.n<~b
_ .(h)n ~ a(n)cos(- 2rmh/m) x
m
n=]
yo(4n
-(mnX')g,x,e(f,x)-rx,dx
_
e(h)Ir ~, a(n)cos(- 2zrnh/m)[i. + I.],
m ,=1
where
i. = n/2 ff Ko ( 41t~--(mnX))g(x)e(f (x) - rx)dx
and
l. = - ff yo( 4n~--(mnX))g(x)e(f (x) - rx)dx.
We first observe that the contribution from the integrals i, is negligible. We have
x/(nMi)/m >>x/nM~ so that
m-1 ~, a(n)ti,l<<m-tG ~ a(n)exp(-A~/nM~)
n=l
n=l
<< G e x p ( -
AM~).
Write the integrals I, in terms of the Hankel functions to get:
l =irb[H(t)(4nx/(nx)~_Hto2)(4n~(mnX))]g(x)e(f(x)_rx)d x
LL~
m /
=i(i)
n
r(2)
-- In
where
I u). = in- 1ml/2n- 1/,*r t' x- 1/4g(x) [ 1 + gi ( 4 7 z ~ n x ) ) ] e(p~,,(x))dx.
~a
Notice that this is same as the integral 'I~ )' in the proof of theorem 3.5 with k = 1.
Similarly for the sum S2(u ) and putting these two terms together we get the
transformation formula claimed in the theorem. Note also that the 'Rankin's trick'
has been extended to the case of Maass forms to get the mean value estimate:
la(n)l 2 =
CX + O(Xa/5+').
n<<.X
We now proceed to give analogs of the above transformation formulae for smoothed
exponential sums provided with weights of the type r/j(n) of pp. 9. We get much better
error terms but we will have to allow for certain weights to appear in the transformed
sum as "well.
Transformation formula for exponential sums
17
Theorem 3.7. Suppose that the assumptions of the theorem 3.5 are satisfied. Let
U >>F - 1Mll +aF1/2 r- 1M~, and J be a fixed positive integer exceeding a certain bound.
Write for j = 1, 2
1 ) J - I J U = M ( r ) + ( - 1)~mj,'
Mj=M~+(-'
and suppose that m'~m~. L e t nj be as before and
n ~ = ( r - f'(M~))2m2 Mj.
Then defining the weights rD(x) in the interval [ M t , M2] as in pp. 9 we have
~D(n)a(n)g(n)e(f(n))
Mt <~n<~M2
2
= A ~ (-- 1)j - 1 ~ wj(n)a'(n)e(nh'/m) x
n<n
j=l
n -(k/2)+
)
1/4X~2)-(3/4)g(Xj,n) X
p'j.n(Xj,n)- l/2 e(Pj,n(Xj,n) + 1/8) +
O(F- t Glhl3/2m -
t/2 Mflk- 1)/2m11/2U L).
where w~(n) = 1 for n < nj, and wj(n) << 1 for n < n'j; further w.i(y) and w~(y) are piecewise
continuous functions in (n'j, nj) with at most J - 1 discontinuities and w~(y) <<(nj - n~)- x
for y~(n'i, nj) whenever w~(y) exists.
[]
The proof of this theorem is the same as that of theorem 3.5 but uses theorem 3.2
in place of theorem 3.1 for details see [6]. A similar theorem holds for the
nonholomorphic case.
A particular case
We now want to specialise the transformation formula to the case of Dirichlet
polynomials, that is to say, to
S(M t, M2 ) =
~
a(n)-(k/2) - it
Ml <~n<~M2
when M 1 < t/2nr < M 2 with r satisfying the conditions of theorem 3.5 and where
a(n)'s are Fourier coefficients of a cusp form of weight k. Such sums occur, for instance,
while estimating the Dirichlet series (associated to cusp forms) on the critical line
and studying their zeros on the critical line.
Here g(z) = 2 -k/2, f(z) = - (t/2n)logz and M ( - r) = t/2~r. The assumptions of the
theorems 3.5 are satisfied (with - r in place of r) if we choose F = t and
Ca= M;k/2rk/2t -k/2. Then n~ = h 2 m ] M f 1, Mi = (t/2rr) + ( - 1)imj and the function
p~,,(x) takes the form
pj.,(x) = - (t/21r)log x + rx + ( - 1)i - l(2~/ (nx)/~ - (k - I)/4 - 1/8)
where 9 = m if (m, N) = N and ~ = mx/N if (m, N) = 1. Assume for sake of simplicity
18
C S Yooananda
that (m, N ) = N; the other case is entirely similar. Thus x i,. ' s are the roots of the
equation
p'j,n(X) = -- t/2nx + r + ( - 1)i-lx/n(mx/x) - t = 0
or equivalently of the quadratic equation
X2
-
-
((t/Irr) + (n/h2))x + (t/2lrr) 2 = O.
Therefore, since x x,. < X2,n, We have
t
n
(-- 1 ) J ( n 2
h k n t ) 1/2
and
t
n
(t/2nr)2xf'~ = 2~r + f~2
(-1)i(_~
hknt~ 1/2
-h~
+ 2;t ] "
To wirte the transformation formula here we need to calculate 2-U2m-1/2x-3/4p,,
j,n
j,n
(x j..)-m and pj,,,(xj,,,).
We have
p j,.(xj..) - t/2rcxi. . + ( - 1)J2 - I nl/2m- 1X-a/2j.n
pt
--
2
So
2mXj,n
3/2 Pj,n(
,, X j,n) = 7 t - l m t x j,n
-1/2 d- ( - - 1)in 1/2
= ( _ 1)1- xh2n - u2(2(t/21tr)2x1-t _ t/~tr) + ( - 1)in 1/2
= It-1/2(2hkt)l/2
i/2
1 + 2-~tt) "
Thus
2-1/2m- 1/2x~f/4p~,.(xi,.)- 1/2 = rcl/4(2hkt)- x/4 1 +
Calculation of p~,.(xi,.) is more delicate. We have
(2~rt- ~xj,.) (-
TM =
--
1 + ~t +
2-~
\hkt]
+
+ hkt ]
1+2---~)
)
whence l o g ( 2 ~ r t - i x j.,) = ( - 1)J2 arcsinh ((~n/2hkt) t/2). We also have, by p~.,(xi.,) = O,
2nrxj,. + 4n( - 1)J- x n 1/2 x;,.1/2m - 1 = 2t - 2nrxi, .
/ 7gyl \ 2~, 1/2
= t - h--k+ ( - 1)i-
\2hkt
~t
"
Thus
27rpi,.(x,,.)=(_l),_~(2t~(~n ~
\
\2hkt/
~(k-l)
2
4 ) -- t log ( ~ )
-
+
7ttl
+ tlogr + t - - ~ .
Transformation formula for exponential sums
19
where we have put ~b(x) = arcsinh (x 1/2) + (x + x2) 112. Thus we have
e(pj..(xj..+l/8)
e(-n/2hk)exp(i(-1)J-l(2tq~(
~kt)
n(k-1)
2
4) X
rUexp (i(t + n/4))(2r~/t)a
Thus finally we have:
S(M1, M2) =
~
a(n)n-(k/2)-" =
MI<~n<<.M2
= nl/4(2hkt)- 1/4
x E ~a(n)e
j=1.
exp(i(t + n/4))
n
2 k
na/4)-(k/2) 1 +
x exp(i(- 1)j- 1 2t~b
t
2
+ O(hmll/2t- 1/2 L 2) + O(h- 1/4ma/4m~ 1/4 L).
The smoothed version in this case reads:
S(MI, M2) =
~
rl#(n)a(n)n-(~/2)-" =
M I <~n<~M2
__ nl/4(2hkt)- 1/4
exp(i(t + n/4))
xj~l.~<.ja(n)e(n(~
x
(
(nn
2t~b 2 ~ t
2~k))
7t(k-I)4))+O(h2m_lml/2t_a/2UL)"
2
It is advantageous to choose U as small as the condition
U >>F-1/2 M~ +a~F1/2 M11+ar-1, i.e. U~F1/2+~r-1.
With this choice the above error term becomes O(F-l/2+~G(Ihlm)l/2M~k-1)/2m11/2 ).
As usual we have a similar formula for the non-holomorphic case.
Remark In the case of Dirichlet series coming from cusp-forms of higher level, N/> 1,
the point of interest is tx/N/2n, and mi, the length, satisfies: t 1/2 +6<< ml <<t. We can
manage to get the same transformation formulae taking M(r) = tx/N/2n where r is
20
C S Yooananda
an approximation to ~/N which satisfies:
Ir - 1/~/NI << t 1/2, r = h/m, m << t 1/4
with (m, N ) = 1.
It can be verified that the order of nj remains unaltered and so will other estimates
which depended on f ' ( M ( r ) ) = r. For example let us look at I - r - f ' ( M t ) [ :
M, = tx/N/2n - mx= tx/N/2~(1 - 2nmx/tx/N);
So
M ~ x = 2rc/tx/N(1 - 2 ~ m l / t x / N ) -1 "" 2rc/tx/N(1 + 2rim1/tx/N) as m 1 <<t I -a;
and f ' ( M 1 ) = - t/2rcM l . Thus
I-r-f'(Mx)l
= I r - 1/x/N0 + 2 n m a / t x / N ) l
= ]r - 1 / x / N - 2rim 1 / N t ],.V-,ma t - x
x ( = m I F M - ~ 2, as F = MI = t).
We will make use of this remark in our application to 'zeros on the critical line' in
the next section.
4. Applications
In this section we give two applications of the transformation formula. The first
application deals with the zeros on the critical line of the Dirichlet series ~b(s)associated
with holomorphic cusp forms and the second application deals with the order of
4)(k\2 + it). In all these applications we use only Rankin's meanvalue estimate though
in the ease of holomorphic forms the estimate a(n) <<n(~- 1)/2+, (Ramanujan - Petersson
conjecture) is known due to Deligne. Thus these results go through in the case of
Maass forms as well where the analogue of Rankin's estimate has been proved but
Deligne's estimate has not yet been; the best result known here is a ( n ) = O(n t/s+')
due to Serre.
Z e r o s on the critical line
Consider the Dirichlet series dp(s) = Z a ( n ) n -s where a(n)'s are Fourier coefficients of
a cusp form of weight k, level N and character e; this series satisfies the following
functional equation
(2n/x/N)-'F(s)~(s)
= C ( 2 ~ / x / N ) ~ - ~ F ( k - s)qJ(k - s),
where [C[ = 1 (for a proof take m = 1 in theorem 2.1 of w If e is a real character
then f ~ f l n ( N ) is an automorphism of M ( N , k, ~) and since it is an involution we can
decompose M ( N , k, e) further as M + (N, k, e) + M - (N, k, e) where on M + (N, k, e ) H ( N )
acts by _+ 1. Thus if f e M • (N, k, e) then b ( n ) = + a(n) in the earlier notation. In this
situation if we rewrite the functional equation as
d?(s) = C'A(s)d?(k - s),
A(s) = (2n/~/N)2s-hr(k - s)/r(s), C' = + C
Transformation formula for exponential sums
21
and further assume that a(n)'s are real we see that on the critical line A(s) has absolute
value 1, IA((k/2)+ it)l = 1. Therefore the function
Z~(t) = [C'A((k/2) + it))]-t/2 ~b((k/2) + it)
is a real function of t. We can now use this function to check whether O(s) has any
zeros on the critical line for t in an interval I T - H, T + / - / ] by comparing the integrals
IE
I
Z ~ ( T + u)du and
IZr
u)ldu
-H
for if ~(s) does not vanish for t in the above interval then these two integrals should
coincide.
Theorem 4.1. Suppose that a(n) is real for all n. Then for all e > 0 there exists a number
To = To(e) such that for all T>. To the function cp(s) has a zero (k/2 + iy) with
I T - ~1 <~ T 1/3+~. A similar statement holds for the function dp(s, l/N).
[]
Proof. We shall first prove the theorem for the function O(s, l/N). Observe that for
the Dirichlet series c~(s, 1/N) also the Corresponding function
Z , (t) = [CA(k~2) + it)] - l/2dp((k/2) + it, 1/N)
is real by virtue of the functional equation proved in theorem 2.1. Also note that
here e need not be a real character and that the result is true for ~(s, h/m) where h
is such that h 2 - - 1 (mad m). Suppose that d~(k/2 + it, 1/N) does not vanish for t in the
interval IT - H, T + HI. Then Zr
is of constant sign in the above interval. Let
H = T (1/a)+3~ and consider the integral
I=
f-
Z , ( T + u ) e x p ( - (u/Ho)2)du,
-1t
where H o = T 1/3+2~
It is well known that
[Z,(T+ u)lexp(--(u/Ho)2)du
Ill =
-H
>>
i
[Zr
u)ldu >>H o
-Ho
See Theorem 3 in [1] for a proof.
We shall estimate I in a different way by making use of the following representation
for dp(s, 1/N) on the critical line:
Lemma 4.2 Let t >12 and t 2 <<X <<t A where A is an arbitrary positive constant. Then we
have, putting a' (n) = a(n)e(1/N),
O(k/2) + it, l/N) = ~ a'(n)n -(k/2)-" +
n<~ X
+ (log 2)- 1
~'
a'(n)log(2X/n)n-(k/2}-it + O ( t X - 1).
X <n<~ 2 X
[]
22
C S Yogananda
Proof. The proof is standard (see for example [6]).
Take X = T 3 and let KE[T 2/3-~, 2T2/a-e]. We have
I=
~
a,(n)n-(k/2)-ir C - 1/2
n<~ T 3
I n - TN/21tl > K
A(k/2 + i(r+ u))- 1/2n-i"exp(- (u/fto)2)du
x
-H
H
+
2
+ exp(-
(u/Ho)2)du + (log 2)- x
~
a'(n)log(2T3/n)n -(~/2)-ir x
T 3 <n<~ 2 T 3
A(k/2 + i(r+ u))- 1~2n-i"e x p ( - (u/Ho)2)du + 0(1)
x C - t/2
d-H
=11 + I 2 + I a + 0(1).
We will now show that Ix and 13 are small. Let first n > TN/2zt + K, and estimate
the integral,
f
H A(k/2 + i(T+ u))-t/2n-iUexp(-(u/Ho)2)du,
-1t
by looking at the corresponding complex integral over the rectangular contour
with vertices + H, + H - i H o. By Sterling's formula we have (remember A(s)=
(2~/N) 2 ' - ~r ( k - s)/r (s))
A(k/2 + i(T + u))-1/2n-iU = exp(i( Tlog(TN/2n)
- T+ ulog(TN/2nn) + O(1))).
On the vertical sides this is bounded and
exp(- (u/Ho) 2) <<exp'( - T).
On the horizontal side in the lower half-plane exp(-(u/Ho) 2) is bounded and
A(k/2 + i(T+ u))- 1/Zn-iU<<exp{ - Holog(2nn/NT)} <<e x p ( - ATe).
For n < TN/2~r- K the corresponding integral can be estimated similarly by
integrating in the upper half-plane. Thus I1 and I3 are << 1.
Coming to 12 we have
I2<<H s'up t
IT - t[ ~ H
~
a'(n)n-(k/2)-it I
In - TN/21tl <~ K
<<H sup I
~
I T - t I <~ H [ In - tN/21t[ <~ K
a'(n) n-(k/2)-it +0(HT-1/sO+3~/2)
Transformation formula for exponential sums
23
The error was obtained by Rankin's estimate with error term:
la(n)l 2 = h x ~ § O(xk-2/S).
n~x
We shall estimate the above sum by applying the transformation formula from w3
with r = 1IN and M j = t N / 2 n + ( - 1 ) J K . Then nj<<t 1/3-2~, and the above sum is
<< T-3~/2. Thus
1II <<Ho T -el2.
But this contradicts 111>>Ho if T is sufficiently large. Hence the assertion.
Now, coming to the Dirichlet series q$(s) we have
A(k/2 + it)-1/2 n-iu = exp(i(Tlog(T~/N/2n)- T + u log(Tx/N/2nn) + O(1).
Hence the sum which we will have to estimate will be over an interval around
Tx/N/2n. Here we will have to use the remark made at the end of w Because of
the approximation of x/N by r = him we will have to apply the smoothed version of
the transformation formula. So instead of the integral I above we will start with its
smoothed version Ij
11 =
~ j ( r + u)Z~o(r+ u ) e x p ( - (u/Ho)2)du.
-H
As in the previous case we have Ilal >>no.
Proceeding as before but breaking the sum at I n - T ~ / N / 2 n l <<,K-v where
v = v I + v 2 + ... + vs is the smoothing parameter, we get
I s = l ' 1 +12 + l a +0(1),
where now
I2' =
f-
-n
[C'A((k/2) + i ( r + u))- l/2~s(r+ u)
x(
a(n)n-k/2-"r+'})exp(-(u/Ho)2)du.
~
tn - T x / N / 2 n l <~K - v
Thus
I
II~l<<n
sup
[
E
rlj(n) a(n)n -k/2 -it .
IT-tl~< H I I n - T x / N / 2 n l < ~ K - v
Now estimating as in the previous case but now using the remark at the end of w3
and smoothed version of the transformation formula we conclude that the above
sum is << T -3't2 and so I z is <<Ho T-~/z. The integrals 11 and 13 are estimated as
before.
r
Estimation of 'long' sums and order of dp(k/2 + it)
Here we are concerned with exponential sums
~,
M <~n <~ M '
a(n)#(n)e(f(n))
t
#
24
C S Yogananda
which are "long" in the sense that the length may be of the order of M itself. It is
not practical to transform such sums directly as in w because variations in f'(x)
might be too much in the interval [M, M']. It is advisable to first partition [M, M ' ]
into segments such that f ' ( x ) practically remains a constant in each segment and
then transform these short sums. But we need to assume that f'(x) is approximately
a power to be able to get some saving in the estimate. The precise result (theorem 4.6
in [6]) is as follows:
Theorem 4.3. Let 2 <<.M < M' <~2M and let f be a holomorphic function in the domain
D = {zl Iz - xl < cM for some x ~ [ M , M'] } where c is a positive constant. Suppose that
f ( x ) is real in [M, M'] and that either
f(z) = Bz~(1 + O(F- ~/3)), zeD
where ~ ~ O, I is a fixed real number and
F = IBIM ~
or
f ( z ) = Blogz(1 + O(F-X/3)),
z~D with F = IBI.
Let g 6 C I [ M , M '] and suppose that for x 6 [ M , M ' ]
Ig(x)l <<G,
Ig'(x)l << G'.
Assume further that M 3/* <<F <<M 3/2. Then
ME n~<~U'a(n)n-~k- l~/2g(n)e(f(n)) <<(G + M G , ) M l l 2 F l l 3 +~
where a(n)'s are Fourier coefficients of a cusp form.
[]
We will not give a proof here since Jutila's proof for the full modular group case
goes through except that a slight modification is required since (unlike in that case
in our situation) we do not have transformation formulae for M~ < t/2nr < M2, where
r(= h/m) is a rational number, for all r; we need to assume that (m, N) = 1 or N (N
is the level of the cusp form) to get a transformation formula. The required modification
is as follows: Put Mo = F 2/3+6 and let K = (M/Mo) ~/2. We may suppose that M/> Mo
for, otherwise the assertion is trivial. Consider the Farey sequence of order K and
drop all those fractions him with (m, N ) > 1. Denote this set of fractions by K. If
r = him and r ' = h'/m' are two consecutive fractions in K let p = (h + h')/(m + m') be
their 'mediant'. We have
p - r = (mh' - m'h)/m(m + m')
In the usual case we would have p - r = 1/m(m + m'); but order-wise both are same
i.e. 1~inK. Define the points M(p) by f'(M(p)) = p and break the given sum at points
M(p) lying in the interval [M, M']. The rest of the proof is as in [6].
COROLLARY 4.4
We have
I~b(k/2 4- it)l <<(Itl 4- 1)x/3+'.
[]
Transformation formula for exponential sums
25
Proof. We have the following approximate functional equation for ~b(s), for 0 ~<a ~<k
and t >/10:
c~(s) = ~ a(n)n -s + ~(s) ~ b(n)n s-k + O(xk/2-'logt)
n<~ x
a~<y
where
x , y >. I,
x y = (tx/N/2~) z and r
= (2rc/x/N)2"-kF(k - s)/F(s).
This reduces the proof of the corollary showing that for all (positive and negative)
large values oft and for all M, M' with 1 ~ M < M' <. tx/N/2rc and M' ~<2M we have
I M < ~ . ~ . a ( n ) n - ~ / 2 - i ' [ <<t '/3+e.
This is precisely the estimate of the theorem 4.3 applied to this sum.
Acknowledgements
The results appearing in this paper formed the contents of the author's thesis submitted
to the Madras University. He wishes to thank his thesis advisor R. Balasubramanian
for his help and encouragement. The author would like to thank M. Jutila for many
useful suggestions. Thanks are also due to D. Prasad and Kirti Joshi for their keen
interest in these results and many helpful discussions. The author would also like to
thank the referee for printing out certain mistakes in an earlier version of the paper.
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