SUMS OF FOURIER COEFFICIENTS OF AUTOMORPHIC CUSP FORMS

by
Theran Bassett
B.Sc., University of Northern British Columbia, 2019

THESIS SUBMITTED IN PARTIAL FULFILLMENT OF
THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE
IN
MATHEMATICS

UNIVERSITY OF NORTHERN BRITISH COLUMBIA

April 2022
© Theran Bassett, 2022

Abstract
Let f be a primitive cusp form of weight k and level N . Let L ( s, sym2 f ) be the

symmetric square L-function associated to f . We denote by Asym 2 f ( n ) the n-th coef ficient in the Dirichlet series representation of L ( s, sym2 f ) . The work in this thesis
is motivated by the relation between the size of partial sums of the coefficients

and the first negative coefficient in the sequence {Asym2 f ( n )}n . We prove that for x

large enough, if Asym2 f ( n )

0 for all n

x, then

Y_ Asym ( n) cNx ( logx ) for
2f

2

n x

^

( n,N ) = 1

some positive constant CM depending only on N . Assuming the Generalized Riemann Hypothesis, we also determine a range of x, in terms of k and N , for which
n $:x

Asym 2 f ( n ) = o ( x ( logx ) 2 ) . These results are GL3 analogues of recent work of

Y. Lamzouri on "Large Sums of Hecke Eigenvalues of Holomorphic Cusp Forms".

We also extend the work of Lamzouri to the number field setting by proving analo-

gous results for the sequence {Af ( ti )}„ of normalized Fourier coefficients associated
with a primitive Hilbert cusp form f defined over a totally real number field of
narrow class number 1.

ii

TABLE OF CONTENTS

Abstract

ii

Table of Contents

iii

Acknowledgements

iv

1 Introduction
1.1
Motivation
1.2
Research Problem
1.3 Thesis Overview
1.4 Conventions and Notation

1
1
3

4
5

Classical Modular Forms
2.1 Definitions and Basic Properties of Modular Forms
2.2
L-functions and Symmetric Square L-functions . .

6
16

3

Review of Literature

21

4

Sums of Coefficients of Symmetric Square L-functions
4.1 Analytic Tools and Preliminary Lemmas
4.2 Proof of Theorem 4.0.1
4.3 Proof of Theorem 4.0.2

25
26
29

36

5

Sums of Fourier Coefficients of Hilbert Modular Forms
5.1 Definition and Basic Properties of Hilbert Modular Forms
Proof of Theorem 5.1.2
5.2

56
56
61

6

Concluding Remarks

76

2

Bibliography

6

76

iii

Acknowledgements
First, I would like to begin by expressing my profound appreciation to my bril-

liant supervisor, Dr. Alia Hamieh. The past few years have been a turbulent time
for myself and much of the world . One of the few constants throughout the duration of this thesis, was the consistent readiness of Dr. Hamieh to provide me with

her invaluable insight or much needed encouragement . Due to her commitment

to being a wonderful supervisor, I have learned priceless skills that will forever

benefit me as a mathematician and as a person. I would also like to extend my
sincere gratitude towards my committee members, Dr. Edward Dobrowolski, and

Dr. Margot Mandy for their very valuable time and advice. For the former, were it
not for his passion of mathematics and engaging teaching style, I would not have

pursued mathematics. So a special thank you to Dr. Dobrowolski for bringing me
into the world of mathematics.

In addition, I would like to thank the entire Department of Mathematics and
Statistics at UNBC. Working alongside its members was always easy due to their

friendly natures and bountiful advice. Of course, my sincere gratitude goes to my
biggest supporter and anchor, my wonderful wife.

iv

Chapter 1
Introduction
1.1 Motivation
There is a quote attributed to the German number theorist Martin Eichler which

says "There are five fundamental operations of mathematics: addition, subtraction, multiplication, division, and modular forms". This bold statement conveys

the importance Eichler, and incidentally many other mathematicians, place on the

topic of modular forms. Modular forms appear in many areas of mathematics

including algebraic geometry, representation theory and number theory as well as
areas outside of pure mathematics with applications to string theory [20]. This field

of research is most closely associated with number theory since many important
arithmetic results have been derived from the theory of modular forms. Most no-

tably, Andrew Wiles proved Fermat's Last Theorem by connecting modular forms
to elliptic curves. Within the short history of the theory, many mathematicians
have been allured to researching modular forms. This is owed to the potential

strength of results that lie within the theory in addition to the broad spectrum of
mathematical techniques that are used to study modular forms. In this work, we

explore questions related to the Fourier coefficients of modular forms from an ana-

1

lytic perspective. Interest in the sign changes of these coefficients, and in particular
the first sign change, spiked when Kowalski, Lau, Soundarajan, and Wu [15] made
a strong connection between this problem and the problem of least quadratic non-

residue, which is a classical topic in analytic number theory.
The Fourier coefficients of modular forms encode a lot of arithmetic information and are worthy of deep research. By just scratching the surface, one can show

several deep results. As an example, Jacobi's 4-square problem asks how many

ways an integer can be written as the sum of four squares. For instance, there are
eight ways to represent 1, since we count all the ordered 4-tuples ( a, b, c, d ) such
that a2 + b2 + c2 + d2 = 1 . Jacobi gave a wonderful elementary answer by inves-

tigating the coefficients of a famous modular form, the theta function. The theta
function is defined as the infinite sum
00

0 ( z) =

X
neZ

qn2 = 1 +

2

e2mn " =
neZ

2 qn,

where q = e 2mz , Im ( z ) > 0

n=1

and is known to be a modular form of weight

\.

Jacobi noted that the n-th Fourier coefficients of 04 ( z ) , which is a modular form
of weight 2, would be exactly the number of ways n can be represented as the sum
of four squares since

04 ( z ) =

Y_ qn Y- qm Z
2

2

neZ

meZ

TGZ

=
SGZ

Y
, ,r ,seZ

nm

qn2+m2+r2+s 2 = 1 +

00

^

r4 ( u ) qn .

n =1

Jacobi then had the task of finding an expression for the theta function from which
he could easily deduce the coefficients of 04 ( z ) . He accomplished his task by writing the theta function in terms of another classical modular form, the eta function.

2

From there it was simple to establish the incredible formula

r4 ( n ) =

SY_ d,
d |n

|
d4

where r4 ( n ) represents the number of ways we can write n as a sum of four squares.
Other less obvious arithmetic properties can be deduced from understanding

coefficients of modular forms. Let

dk

ffk ( n ) =
d|n

be the sum of the k-th powers of the positive divisors of n. If asked to prove the
relation
TL — 1

<T7 ( H ) = cr3 ( n ) + 120

^

03 ( m ) o 3 ( H — m ),

(1.1)

one would likely delve into a deep confusing mess of sums and divisors. It cer-

tainly would seem like there is no short and simple proof of this, but there is! In
fact, all that is required for its proof are standard facts about modular forms that we
mention in Chapter 2. We present a proof of this result after formally introducing

the theory of modular forms.

1.2

Research Problem

In this thesis, we study sequences formed by Fourier coefficients associated with
modular forms. To this end, we prove three main theorems that are extensions

of recent work of Y. Lamzouri in [16]. Let f be a primitive cusp form of weight
k and level N . Let L ( s, sym2 f ) be the symmetric square L-function associated to
f . We denote by Asym2 f fu ) the n-th coefficient in the Dirichlet series representa-

3

tion of L ( s, sym2 f ) . Our work is motivated by the connection between the size of

Asym
^
following
prove

sums of the form
{\ ym2 f ( rL )}n - We

nrx

2f

( u ) and the first negative coefficient in the sequence

the

theorems. The reader is referred to Chapter 2

for the relevant notation and terminology.

Theorem 1.2.1. Let e > 0 be given, and letx

Suppose that Asym 2 f ( n )

0 /or all n

^

^

l be such that logx > max o> ( N ) 1 + e,

x. Then

2

Y- Asym ( n ) cNx (logx ) ,
2f

nsjx

( n,N ) =1

where cN is a positive constant depending only on N .

We also prove that the sum appearing in Theorem 1.2.1 is o (x ( logx ) 2 ) in a certain range of x depending on k and N .

Theorem 1.2.2. Assume that L ( s, sym2 f ) satisfies the Generalized Riemann Hypothesis .
*
In the range loglog
log kN

—y oo, we have
HAsym fW = 0 (x ( logx ) 2) .
n x
^
2

These theorems can be viewed as GL3 analogues of [16, Theorem 1.1] and [16,

Corollary 1.2]. We also extend this work to the number field setting by proving the
analogue of Theorem 1.2.2 for the sequence {Af ( n )}„ of normalized Fourier coefficients associated with a primitive Hilbert cusp form f defined over a totally real

number field of narrow class number 1.

1.3 Thesis Overview
This thesis is organized into six chapters. In Chapter 2, we introduce the basic

theory for modular forms and their associated L-functions and symmetric square
4

L-functions. In Chapter 3 we briefly present the major contributions leading up

to the work of Lamzouri [16] which inspired this research project. Our aim in

this chapter is to focus on implications of the results, the methods of their proofs,
and the constraints the authors worked within, rather than providing a rigorous

mathematical account. In Chapter 4, we prove Theorem 1.2.1 and Theorem 1.2.2

after presenting all the required technical results from complex analysis and an-

alytic number theory. In Chapter 5 we provide an overview of Hilbert modular
forms and prove a main theorem related to their Fourier coefficients. This theorem
mirrors Theorem 1.2.2, but in the number field setting. In Chapter 6 we describe

possible avenues for future research.

1.4

Conventions and Notation

In this work, we adopt the following conventions and notation. Given two functions f ( x ) and g ( x ) we write f ( x ) < g ( x ), g ( x ) » f ( x ) orf ( x ) = 0 ( g ( x ) ) to mean there
exists some positive constant M such that |f ( x ) | y M| g ( x ) | for x large enough. We
write f ( x ) x g ( x ) when both estimates f ( x ) C g ( x ) and g ( x ) < f ( x ) hold simultanef (x)
ously. We write f ( x ) ~ g ( x ) if lim —— = 1 . We write f ( x ) = o ( g ( x ) ) when g ( x ) 0
<

^

f fx )

for sufficiently large x and lim —— = 0. We follow the convention of denoting
an arbitrarily small positive constant by e, which may change from instance to instance. The letter p ( resp . p ) will be exclusively used to represent a prime number
( resp . prime ideal) .

5

Chapter 2
Classical Modular Forms
In this chapter, we provide a brief account of the theory of modular forms. The
reader is referred to standard textbooks on the topic such as [3] or [11] for a com-

plete exposition. All the theorems discussed in this chapter can be found in the
aforementioned references.

2.1

Definitions and Basic Properties of Modular Forms

Definition (Upper Half Plane ). The upper half plane, which we denote by H, is
the set of complex numbers with positive imaginary part. More precisely, we have

H = (z e C : Im ( z ) > 0}.
Let S1_2 ( Z ) be the group of 2-by-2 matrices with integer entries and determinant
1. This group is referred to as the modular group. For y e S1_ 2 ( Z ) and z e H, we

set
az + b

where y =

a b
c d

6

Observe that for z = x + iy e H and y e SL2 ( Z ), we have
az + b _ a ( x + ty ) + b _ ( ax + b + lay ) ( cz + d — icy )
( cx + d ] 2 + ( cy ) 2
cz + d
c ( x + iy ) + d

and so

Im ( yz ) =

V
( cx + d ) 2 + ( cy ) 2

> 0.

Hence, yz e H for all z e H and y e SL2 ( Z ) .

Definition (Modular Transformation). Let k be a positive integer. Given a function
f : H -> <C, we say that f satisfies the modular transformation property of weight k

if
f ( yz ) = ( cz + d ) kf ( z ) ,

Vy = ( *

\)

SL2 ( Z ) .

(2.1)

In particular, (2.1) implies that f is periodic of period 1 since

f (z + l ) = f

((

oi

) z) = 0 ) kf ( z) = f ( z ) .

Similarly, when (2.1) is applied with y = (

01

) , we see that f = 0 unless k is an

even integer. Henceforth, we will assume that k is even.

Definition (Modular Form ). Let f : H -> C be a function that satisfies (2.1) . We say
that f is a modular form of weight k if , in addition, it is holomorphic on H and at
oo .

Next, we explain the requirement that f ( z ) is holomorphic at z = oo . The map g
defined by z ^ e 2niz transforms H to the punctured open unit disk D * = {q e C :
0 < | q | < 1 ). Since f is periodic of period 1 , we can write f ( z ) = g ( e2mz ), where g ( q )

7

is holomorphic on D * . The function g has a Laurent series expansion

g( q) =

Y
- a( ^ cin
TIEZ
rL

_

for q e D * . Since | q | = e 27lIm ( z ) , we see that q

0 as Im ( z )

oo. Hence, thinking

of z = oo as lying far in the imaginary direction, we say that f is holomorphic at oo
if g extends holomorphically to the puncture point q = 0 in which case its Laurent
series sums over n e N .

This means that a modular form f has a Fourier series expansion
OO

t( z) =

22

n=0

where q = e 2mz . The coefficients a ( n ) are called the Fourier coefficients of the

—

modular form f . Showing that a holomorphic function f : H » C that satisfies
(2.1) is holomorphic at oo doesn't require computing the Fourier expansion of f . In
view of the relation q —> 0

Im ( z )

oo, it suffices to show that

lim

Im ( z)-Kx>

f (z)

exists.

Definition (Cusp Form ). A cusp form of weight k is a modular form f of weight k
whose Fourier expansion has leading coefficient a ( 0 ) = 0, i.e.
OO

f ( z) =

^

a ( u ) qn,

q = e2niz .

n=1

It is well-known that the set of all modular forms of weight k forms a finite
dimensional vector space over C denoted by Mk ( SL2 ( Z ) ) or simply Mk . The set

of all cusp forms of weight k forms a subspace of Mk, denoted by Sk ( SL2 ( Z ) ) or

8

simply Sk . The dimension of Mk as a vector space over C is given by

dim ( Mk ) =

if k = 2 ( mod 12 )
LfsJ
[£ J + 1 if k 2 ( mod 12 ) .
^

In particular, it easy to see that M2 = {0} and that Mk is one dimensional for all
even integers 2 < k

10. Let k > 2 be an even integer. For non-trivial examples of

modular forms in Mk, consider the Eisenstein series Gk given by
1

Gk ( z ) =
( m,n ) y ( 0,0 )

( mz + ri ) k '

Z

H.

This can be viewed as a 2-dimensional analogue of the Riemann zeta function
00

C( k ) =

1

One can show that Gk is indeed a modular form of weight k. More-

n=1

over, it admits the Fourier expansion

Gk ( z ) = 2 C ( k ) + 2 -

( 2m ) k

^

In fact, for k e N, one can prove that any f e Mk can be written as a polynomial in
G 4 and Gg . In practice, it is common to work with the normalized Eisenstein series

of weight k

GkW

F M]

2C ( k) '

for which the leading Fourier coefficient is 1. The space of cusp forms Sk together
with the space EkC gives Mk = Sk © EkC if k > 2. In particular, Mg = EgC since
we know that it is 1-dimensional. This observation is at the heart of the proof of

(1.1) presented in Chapter 1. Observe that |
E e Mg, and so Eg = |
E . Hence, the

identity in (1.1) follows by comparing Fourier coefficients. This simple proof gives
us a glimpse to the infinitely many other arithmetic facts that we can derive from
Eisenstein's series alone.

9

In what follows we focus our discussion on the space of cusp forms Sk . In order
to study this space further, we make it into an inner product space.

Definition (Petersson Inner Product ). The Petersson inner product ( , ) : Sk x Sk
C is given by

( f, g ) =

]J ykf z g z dxda
a '
( ) ( )

2

SL 2 ( Z ) \ H

where z = x + iy .
One can verify that the integral appearing in the above definition is well-defined

because the integrand is SL2 fZ ) -invariant. Moreover, the integral is finite since
both f and g are cusp forms. In particular, the space Sk is a finite dimensional
Hilbert space with this inner product.

We now introduce the Hecke operators Tn on the linear space Mk of modular

forms of weight k. Let M ( m ) be the set of all 2-by-2 matrices with integer entries
and determinant m. The modular group SL2 ( Z ) acts on M ( m ) from both sides so

that M ( m ) = SL2 ( Z ) M ( m ) = M ( m ) SL2 ( Z ) . The collection
a b

A ( m) =

: ad = m, 0

b< d

0 d

forms a complete set of representatives of M ( m ) modulo SL2 ( Z ), i.e. we have the

disjoint partition M ( m ) =

[J SI_2 ( Z ) p. The m-th Hecke operator acting on f e

pGA ( m )

Mk is given by the formula

Tmf = mH

£V

pGA ( m )

_

where we set f |y = det ( p ) 2 ( c z + d ) kf ( yz ) when y — ( £ jj ) . In what follows we

present a formula for Tm that is more suitable for computational purposes.
Definition (Hecke Operator ). The m-th Hecke operator Tm acting on a modular
10

form f G Mk is defined by

Theorem 2.1.1. The Hecke operator Tm is a linear transformation on Mk . It maps a

modular form to a modular form and a cusp form to a cusp form:

Tm : Mk

and

Mk

Tm : Sk

Sk .

Next, we present a result that shows how Tm acts on the Fourier coefficients of
f . Let f be given by the Fourier expansion
00

2=>( u ) e

f (z) =

2

2

(2.2)

.

n 0

Then Tmf is given by the Fourier expansion
OO

Tmf ( z ) =

^

bm ( u ) e2

z,

n=0

where

bm ( u ) =

Y_, dk l a ( r ) -

d| ( m n )

^

(2.3)

This relation is key to studying the coefficients of modular forms because it allows
the Fourier coefficients to be expressed in terms of the eigenvalues of the operator.
This is made possible by the following important result.

Theorem 2.1.2. In the space Sk of cusp forms of weight k there exists an orthogonal basis
which consists of eigenvectors of all the Hecke operators Tm . Since each such vector is a
modular form, we refer to it as a Hecke eigenform instead .
Theorem 2.1.2 follows from the spectral theorem of linear algebra because
11

Theorem 2.1.3. The Hecke operators commute, i.e. TmTn = TnTm . Moreover, the Hecke

operators acting on Sk are self-adjoint , i .e. (Tnf , g ) = (f , Tng ) for all f , g e Sk .
Another important result about Hecke operators is that they satisfy the multi-

plicative property Tmri = TmTn for all ( m., n ) = 1 . Moreover, they satisfy the recur_
rence formula Tpj + i = TpTpl - pk 1 Tpj _ i for all j

1.

Let f be a Hecke eigenform satisfying

Tmf = A ( m ) f VmeN .
Suppose that f is given by the Fourier expansion (2.2). Using (2.3), we see that
A ( m) a ( n ) =

^

dk 1 a ( mnd 2 ) .

d | ( ra,n )

For n = 1, this yields A ( m ) a ( l ) = a ( m ) . Notice that a ( l ) f 0 for otherwise f
vanishes identically. Hence, the Fourier coefficients of a Hecke eigenform are pro-

portional to the eigenvalues of the Hecke operators. We say that f is normalized
if a ( l ) = 1 . The set of all normalized Hecke eigenforms of weight k is commonly
denoted by dfk .
The entire theory that we have presented so far is centered around a function
f : H

C that is holomorphic on H and at oo and satisfies the transformation

property
f ( yz ) = ( cz + d ) kf ( z ) ,

Vy = ( “ * ) e SL2 ( Z ) .

Replacing the modular group SL2 ( Z ) by a subgroup F in the last condition generalizes the notion of modular forms and allows for more examples. In fact, the
modular form 0 that was mentioned in relation to Jacobi's 4-square problem in

Chapter 1 is one such example.
In what follows we will revisit most of the above constructions when SLifZ j is
12

replaced by a certain subgroup. The exposition, however, will be less detailed than
the previous.
Definition (Modular Transformation w.r.t F). Let k be a positive integer, and let F

—

be a finite index subgroup of SL2 ( Z ) . Given a function f : H » C, we say that f

satisfies the modular transformation property of weight k with respect to F if
f ( yz ) = ( cz + d ) kf ( z ) ,

When (2.4) is applied with y = ( Q1

^

Vy = ( “

^

) er .

(2.4)

) , we see that f = 0 unless k is an even

integer. Since V has a finite index in SL2 ( Z ), there exists M y 1 such that ( J ' ) e h

Y

In particular, (2.4) implies that

f (z + M) = f

( ( J Y ) ) = C, ) kf (z) = f ( z) .
Z

Hence, f can be written as a function in

which we denote by g . More

precisely, there is a function g such that f ( z ) = g ( qjvi ) . The function g is holomor-

phic in the punctured unit disk. If g extends to a holomorphic function at 0, we say
that f is holomorphic at oo. This gives rise to the Fourier series expansion
OO

f (z) = g ( q M ) =

Y=-

n 0

where qM = e

2

^ . We say that f vanishes at infinity if a ( 0 ) = 0.

If a G SL2 ( Z ), then a-1 Fa is a finite index subgroup of SL2 ( Z ) . For y G F, one
can verify that f a satisfies the modular transformation property for the subgroup
a-1 Fa. Hence, f |a has a Fourier expansion at oo. We say that f is holomorphic at

the cusps if f | is holomorphic at oo for all a G SL2 ( Z ) . We say that f vanishes at the
CT

cusps if f | vanishes at oo for all cr G SL2 ( Z ) .
ff

13

The most commonly used subgroups V are of the form

r0 ( N ) = { ( “ § ) G SL2 ( Z ) : c = 0 ( mod N ) j
and

r, ( IM ) = { ( “ S ) e S L2 ( Z ) : c = 0 ( m o d N )

/

a = d = 1 (m o d N )

}

for positive integers N . Notice for example that Fo ( 1 ) = SL2 ( Z ) is the full modular group. In this chapter we will only discuss modular forms arising from the

subgroups r0 ( N ) .
Definition (Modular Form of Level N ). Let f : H

C be a function that satisfies

(2.4) for Fo ( N ) . We say that f is a modular form of weight k and level N if, in

addition, it is holomorphic on H and at the cusps.

Definition (Cusp Form of Level N ). A cusp form of weight k and level N is a
modular form f of weight k and level N which vanishes at the cusps.

It is well-known that the set of all modular forms of weight k and level N forms
a finite dimensional vector space over C denoted by Mk ( N ) . The set of all cusp

forms of weight k and level N forms a subspace of Mk ( N ) , denoted by Sk ( N ) .
There is a well-developed theory of Eisenstein series of weight k and non-trivial
level N , but we will not discuss it here. In what follows we focus our discussion
on the space of cusp forms Sk ( N ) . In order to study this space further, we make it
into an inner product space.

Definition (Petersson Inner Product ). The Petersson inner product ( , ) : Sk x Sk
C is given by

<

f , 9) =

y k n z ) g ( z ) dxdy '
|
J
D
2

r0 ( N ) \ H

where z = x + iy .
14

Much like the case of Sk, the space Sk ( N ) is a finite dimensional Hilbert space
with this inner product .

One can also define Hecke operators on Sk ( N ) much like we did for Sk . For

economy of exposition, we will not go into the details of this construction. Instead,
we will state the most important theorems that result from this construction.

Theorem 2.1.4. The Hecke operators commute , i.e. TmTn = TnTm . Moreover, for all
f , g G Sk, we have ( Tnf , g ) = ( f , Tng ) if ( n, N ) = 1 .

Therefore,

Theorem 2.1.5. In the space Sk ( N ) there exists an orthogonal basis which consists of

eigenvectors of all the Hecke operators Tn with ( n, N ) = 1 . Since each such vector is a
modular form, we refer to it as a Hecke eigenform instead .
Let f be a Hecke eigenform satisfying

Tmf = A ( m ) f V ( m, N ) = 1 .

(2.5)

Suppose that f is given by the Fourier expansion (2.2). Using a formula similar to
(2.3) for the Fourier expansion of Tmf ( z ) , we get A ( m ) a ( 1 ) = a ( m ) for all ( m, N ) = 1 .

Unfortunately, this doesn't imply that a ( 1 ) f 0, as there are Hecke eigenforms f fO
such that a ( l ) = 0. Such forms are known to come from lower levels and are called
oldforms. In fact, one notices that a modular form is not always unique to its level.
For example, a modular form of level N will necessarily be a modular form of
level M if N |M. When working in the level aspect, it is often useful to ignore those
modular from that come from lower levels.

More precisely, the space of cusp forms Sk ( N ) can be decomposed as Sk ( N ) =
S°ld ( N ) ® S£ew ( N ) where <S £ld ( N ) is the subspace of cusp forms that come from

lower levels. The new space S£ew ( N ) is the orthogonal complement of S°ld ( N ) with

respect to the Petersson inner product.
15

The importance of the new space lies in the following result.

Theorem 2.1.6. Let f e S£ew ( N ) be a Hecke eigenform. Then (2.5) extends to all n. In

particular, a ( l )

^ 0, and so can be normalized so that ( l ) =
a

f

1.

A cusp form f in Sk ( N ) is said to be primitive if it is a normalized Hecke eigenform in the new space. We denote by !K£ ( N ) the (finite) set of all primitive Hecke

eigenforms of weight k and level N .
In light of Theorems 2.1.4 and 2.1.6, we see the Fourier coefficients of a nor-

malized Hecke eigenform f

TCJ ( N ) must be real, since they coincide with the
,

eigenvalues of a self -adjoint operator.

2.2

L-functions and Symmetric Square L-functions

Let f G TC£ ( N ) be a primitive form given by the Fourier expansion

Y=_ af ( n) e
OO

f (z) =

2

z.

n 1

We write af ( n ) = Af ( n ) uV . A simple argument can be used to prove that Af ( n ) <
n. 2 for all n y 1 . One of the deepest theorems in the theory of modular forms is the

upper bound [13, Equation 14.54]
| Af ( n ) KT ( n )

for any n

(2.6)

^ 1 . Here ( ) is the number of positive divisors of n. This bound was
T U

established by P. Deligne [2] in the 1970's, thus settling Ramanujan's conjecture for

classical modular forms.

In order to better understand the asymptotic behaviour of these coefficients
and the many underlying hidden structures, a common approach is to bound their
16

summatory functions over a certain sequence. Standard bounds in this direction
include the following [13]:
*

ZnOclAf ( n ) |2 <f

*

Ln ^ XlAf ( U ) l <f x l°g X,

*

InaAfW «f

^

logW '

for x large enough. In the above asymptotics, we use the notation Sf ( x ] <Cf g ( x ]
C| g ( x ) | for some positive constant C that depends only on

to mean that |Sf ( x ) |

f . Notice the improvement in the bound when removing the absolute value. This

alone may spark the curious mathematician to search for negative coefficients of
modular forms. One of the powerful tools used to improve upon such bounds is
the so-called L-function associated with the form f .
The L-function associated to f is given by the Dirichlet series

£ AfM
n— 1

which is absolutely convergent for Re ( s ) > 1 . It is known that L ( s, f ) admits an an-

alytic continuation to the entire complex plane and satisfies a functional equation
that relates L ( s, f ) to L ( 1 — s, f ) . Moreover, for Re ( s ) > 1, we have the Euler product

representation

+
n ( Af ) ’ n O
Pf pn '
=n(
Ps
’ Ps
i —

u f s] =
/

(p i p s

PIN

Mpjps

p 2s

)

pfN

CCf ( p )

p

-

-1

1

v

17

[

~

(2.7)

where af ( p ) and |3f ( p ) are referred to as the p-th local parameters of f and satisfy

£Xf (

p ) + (3f ( p ) = Af ( p ) , Of ( p ) Pf ( p )

Of ( p ) = ± 7p / Pf ( p ) =

-

1

ifpfN
(2.8)

o

ifp | N .

Since the Ramanujan conjecture for classical modular forms is known, thanks

to Deligne's work, we have | ocf ( p ) | = ||3f ( p ) | = 1 if p \ N . Hence, if p f N , we write
_
0 (p)
and |3 f ( p ) = e 10 f ( p ) for 0 f ( p ) [0, TT] . We also have the relations (see
ccf ( p ) = el f

[19, Equation 1.2, Equation 1.3])
+1 - Mpr +1

Af ( pv ) = MPr( p ) - ( p )
Of
Pf

, (v

^ l)

(2.9)

.

The function L ( s, f ) can be viewed as a generalization of the Riemann zeta function which is given by
oo

=

1

Re ( s ) > 1 ,

n =1

and extends to an analytic function on the complex plane with the exception of
a simple pole at s = 1 . The Riemann Hypothesis is the conjecture that all the

non-trivial zeros of £ ( s ) have Re ( s ) =

\. This is one of the of the most famous

open problems in all of mathematics. It is widely believed that a similar assertion
holds for L-functions of modular forms. More precisely, the Generalized Riemann

Hypothesis (GRH ) for L ( s, f ] is the conjecture that all the non-trivial zeros of L ( s, f ]
have Re ( s ] =

Let us now define the symmetric square L-function associated with a primitive

cusp form f (see [19, Equation 1.4]).
Definition. The symmetric square L-function is given by
L ( s, sym2 f ) =

(1 af (P ) jPf (P ) jP )
Yl
<
“

P 0 j<2

18

2

S

(2.10)

for Re ( s ) > 1 .

Using (2.8) we can write L ( s, sym2 f ) as

2

L ( S sym f ) =
/

Y[ M
p
|N

V

e2i0 f ( p )

1

-

1-

Vs

g — 2i0 f ( p )

Vs

) o r ' n ('
- p-

/

T'

- p- -

'

p |N

(2.11)

We denote by Asym2 f ( n ) the n-th coefficient in the Dirichlet series representation of
L ( s, sym2 f ) so that

L ( s, sym2 f ) =

£

n=1

^

fhl

,

Re ( s ) > 1 .

(2.12)

It is well known that Asym 2 f ( n ) is real and multiplicative. Moreover, we have [19,

Equation 1.6]

| Asym ( -n ) | u (n ),
2f

.

(2.13)

where T3 ( H ) is the n-th coefficient in the Dirichlet series representation of c’ ( s ) .
In fact, x3 ( n ) is the number of ways n can be written as the product of 3 positive

integers. By [17, Eq-1.12] we get

Asym2 f ( P’ ] =
for p|N and ]

sin ( ( ) + 2 ) 9 f ( p ) ) sin ( ( j + 1 ) 8f ( p ) )
sin 0f ( p ) sin ( 20 f ( p ) )

(2.14)

^ 1.

It known that L ( s, sym2 f ) admits analytic continuation to the entire complex

plane and satisfies a functional equation that relates its value at s to its value at
1

s . The reader is referred to [19, Equation 1.9] and the references therein .

The Generalized Riemann Hypothesis for Us, sym2 f ) is the conjecture that the
non-trivial zeros of L ( s, sym2 f ) have Re ( s ) =

19

\.5

A convexity bound for the symmetric square L-functions is given by

(

|L ( s, sym2 f ) | < N 2 ( | S + k| + 3 ) 3

)

4

(2.15)

.

The reader is referred to [13, Equation 5.22] for a detailed exposition on convexity
bounds for automorphic L-functions. Assuming the GRH, we have the follow-

ing much stronger bound which follows from [13, Theorem 5.19] when applied to
L ( s, sym2f ) .

. Assume the GRH for log ( L ( s, sym2 f ) ) .
Theorem 2.2.1. Let s = a + it with \ < a |
Then
2 2g

2 ))

log ( L ( s, sym f

3 (log ( N 2 k2 (|s| + 3] 3 ) )
+ 3 log log N 2 k2 (|s| + 3 ) 3
( 2 cr 1 ) log log ( N 2 k2 ( |s| + 3 ) 3 )

—

with absolute implied constants .

20

^

)

Chapter 3
Review of Literature
The objective of this chapter is to highlight recent contributions pertaining to the
connection between sums of coefficients of modular forms and the first sign change

in these coefficients. Our focus is to analyze the results and to highlight the tools

that the authors employed to achieve their results.

Let f G df £ ( N ) be a primitive cusp form of weight k and level N . Let nf be
the smallest integer n with ( u, N ) = 1 such that the n-th Fourier coefficient of f is

negative. In 2006, Kohnen and Sengupta [14] computed an upper bound for nf in
terms of k and N .

Theorem 3.0.1. Suppose that N is squarefree , and let f e :Kk (' N ) he normalized , then
uf L cikN ( logk ) Aexp

^

c2

y

log ( M + 1 ) \
loglog ( N + 2 ) ) ‘

Here A is any real number > 26, ci is a constant depending only on A, and c2 is an absolute

constant .

In 2007, Kohnen and Sengupta collaborated with Iwaniec [12], seeking to im-

prove their previous bound on the first sign change of a Hecke eigenform. Their
method relies on exploiting Hecke relations. The authors establish the following
21

upper bound for nf .
Theorem 3.0.2. Let f e IH£ ( N j be a normalized Hecke eigenform, then
nf <

where Q = k2 N , and the implied constant is absolute.

Obtaining such a clean result sparked a flurry of activity among researchers
studying modular forms and convinced more authors to investigate sign changes.
Naturally, many sought to answer questions of whether a better bound could be
found, or if constraints on the modular form could be relaxed . In 2010 Kowalski,
Lau, Soundararajan and Wu [15] address the former issue and show the following
theorem.

Theorem 3.0.3. Let f e

) be normalized and let k an even integer

2 and N be any

positive integer. Then
n f < Q 2o ,

( nf , N ) = l .

inhere the implied constant is absolute.

This improvement may not seem substantial, especially considering the authors remark that their result is likely far from optimal. After all, if one assumes
the GRH for L ( s, f ) , it turns out that nf <c ( log kN ) 2 [15]. The most impressive thing
about the work in [15], is the framework they build to prove it . They introduce nu-

merous innovative functions which remain frequently used today. We highlight
once such function below in (3.1). Furthermore, they demonstrate their vast experience across much of analytic number theory by cleverly applying many classical

tools to complex ideas, inspiring new ideas for future authors.

22

Suppose for all u < x with ( u, N ) = 1, we have Af ( TX )

0. The simplest function

introduced for the proof Theorem 3.0.3 is a function defined over prime numbers

pby

— 2 if p > x and p \ N

hx ( p ) =

0

if A/X < p

1

if p

y

x or p| N

(3.1)

/ x and p\ N

and hx ( pv ) = 0 for v > 2.

Under relevant assumptions, hx ( p ) aids the problem of estimating the first sign

change in an interesting way. Given f e 9t£ ( N ) with Fourier expansion f ( z ) =

^

In=i Mn ) n qn , set
Sf ( x; N ) =

Af
n <x
( n,N ) =1

^

'

They show that

Sf ( x; N )

^

hx ( n ),

and proceed to determine a good lower bound for Hn x hx ( ri ) . Then the authors

^

find an upper bound for Sf ( x; N ) using standard techniques. Finally, they com-

pute the desired upper bound for nf by comparing the lower and upper bounds of

Sf ( x; N ) .
Using the same methods, Matomaki [23] improves on [15] by showing a stronger
lower bound for Sf ( x; N ) . She is able to show, under the same assumptions as Theorem 3.0.3, that nf < Qs where the implied constant is absolute. After this result,

improvements on the first sign change of modular forms halted, serving as a testament to the strength of Matomaki's work.
The idea of comparing upper and lower bounds for summatory functions of
Fourier coefficients of modular forms in order to estimate the first sign change was

perhaps a novel idea, but the study of the asymptotic behaviour of such summa23

tory functions over certain sequences of integers was not . In this regard, there is
a rich history dating back to the pioneers of the theory with Hecke showing that

^

Af ( n ) <f xi when f is a normalized Hecke eigenform (see for example [11,

n x

^
5]) . This topic has witnessed a proliferation of interesting results in reChapter
cent years such as [31], [26], [27], [28], [29], [30], [8], [10], [4], [21], [22] and [18] to
mention a few.

Last, but definitely not least, we discuss recent work of Y. Lamzouri on sums

of Fourier coefficients of eigenforms. Motivated by the results and methods of [23]
and [15], Lamzouri [16] proves the following theorem.
Theorem 3.0.4. Let f e "Kk . There exist absolute positive constants b 0, c0 such that if
x

^ bo and A ( n ) 0 for all n x then
Sf ( x ) =

^

A (n)

coxlogx.

n x

^

Paired together with another result from the same paper, Theorem 3.0.4 implies
a sign change bound . This bound is not the best to date, but the method for obtain-

ing it is deserving of interest. Inspired by the work of Granville and Soundararajan
on character sums [7], he then proves the following theorem
y
,
Theorem 3.0.5. Let f 6 Hk , and assume the GRHfor L ( s, f ) . In the range logIQRX
log k — oo

we have

Sf ( x ) = o ( xlogx ) .
Observe how Lamzouri is working only in the weight aspect of Hecke eigenform.

This observation plus the approachable nature of his proofs, were the primary
motivation for this project.

24

Chapter 4
Sums of Coefficients of Symmetric

Square L-functions
In this chapter we prove Theorem 1.2.1 and Theorem 1.2.2. For convenience of the
reader, we restate the theorems below. Recall that cufN ) is the number of distinct

prime divisors of N , and 4> ( N ) counts the number of positive integers less than N
which are relatively prime to N .

Theorem 4.0.1. Let f e 5f £ ( N ) . Let e > 0 be given, and let x

^

max cu ( N ) 1 + e ,

Suppose that Asym2f ( u )

0 for all n

^ 2 be such that logx >

< x. Then

2

Y- Asym fM CNx (logx ) ,
2

n x
( n,N ) =1

^

where

is a positive constant depending only on N .

Theorem 4.0.2. Let f e 5f £ ( N ) , and assume the Generalized Riemann Hypothesis ( GRH )

for L ( s, sym2 f ) . In the range

° *kN ) -> oo, we have
logl|
•

XAsym f ( n ) = o ( x ( logx ) 2 ) .
n x
^
2

25

These theorems provide GL3 analogues of [16, Theorem 1.1] and [16, Corol-

lary 1.2]. Prior to proving Theorem 4.0.1 and Theorem 4.0.2, we present a list of
analytic tools and preliminary lemmas which we use in the proofs.

4.1 Analytic Tools and Preliminary Lemmas
Here we list theorems that are necessary to the proofs of Theorem 4.0.1 and Theorem 4.0.2. Results are stated without proof , but the reader may refer to the pro-

vided reference for the details.

We recall the Borel-Caratheodory Theorem from complex analysis (for a reference, see for example [34, Section 5.5, p. 174]) .

Theorem 4.1.1 (Borel-Caratheodory Theorem). Let f ( z ) be an analytic function on
|z| y R. Let Mf ( r ) and Af ( r ) be the maxima of |f ( z ) | and Re ( f ( z ) ), respectively, on | z| = r.

Then for 0 < r < R , we have

Another standard result from complex analysis that we use extensively in this
work is Perron's formula which relates partial sums of the form

tour integral involving the corresponding Dirichlet series

^fir - an
,

tv

x

to a con-

We need a

truncated version of Perron's formula which can be found in [24, Theorem 5.2]

Theorem 4.1.2 (Truncated Perron's Formula ). Let F ( s ) = Y.n= i yf be the Dirichlet se-

ries associated with a sequence of complex numbers { an}“= 1 , and let aa denote the abscissa

of absolute convergence of F ( s ) . Let x £ Z be a positive real number. If ci = max ( 0, <TQ ),
then
C!

+ix

Xs
F ( s ) — ds + R,
s
ci — ix

26

where
R=

^

_
ansi ( xlog ( n/ x ) )
Y
Y
^si (xloS ( ))
|
< <
< <
Q

“

“

x n 2x

n x

( 4Cl + xCl 2- | an | \

+0
du is the sine integral .

and si ( x ) = — J“

We note here that si ( x ) satisfies si ( x ) C min (1 , 1) and si ( x ) + si ( — x ) = — n for
<

all x > 0.
Next we list some standard results from analytic number theory.

Theorem 4.1.3 (Abel's Summation Formula ). Consider a function f that is continu-

ously differentiable on the interval [y , x] with 0 < y < x. Let a ( n ) be an arithmetic
function, and set A ( x ) =

a ( n ) . Then we have

Y a (n ) f ( n) = A ( x ) f (x ) A (y ) f (y ) f A ( t ) f ' ( t ) dt.
-

-

(4.1 )

tjCruSx

Theorem 4.1.4. [ 24, Theorem 2.7( d ) ] , For x

2 we have

^

)

XNloglogx + b + O
oo

where b = y — Y

"

Y —1 T and y is the Eider-Mascheroni constant .
"

p k =2

kpfc
v

2, we have

Lemma 4.1.5. Let N be a positive integer. For x

Y*
P 5?
pfN

where CN = log

,

“

^

1

= loglogx + CN + O logx

cu ( N )
X

+ y — HP|N Y-\&2 kjA and the implicit constant in the O -term is

absolute.
27

Proof . First, we observe that

z^ 1 = z1 z1
-

P

K
X

p|N

p| N r

p| N K
p >x

PIN K

V

p| N

V

7

.- Llog (, - l) _ ZZ kpk1 + 0 (f )
= - Zi°s
p |N

K /

-

V

z—^ 1 z—^ 1 z—1
L

p

=*

^

(’ ) ZZi + ZZkp +

^

1

-

P/

It follows that

p x r
P
p
|N

p| N k 2

P k 2

^

P

wjc

pfN k 2

^

0

cu ( N )

P

-

p

p x r

p
*
psjx r

PIN

= loglogx + y +

Y_ log ( 1 l ) Y_ Y-

p| N

-

k

pfN k 2

^

(

We get the desired result by observing that Hp | N log l —

^

j = log

j.

Finally, we will need two results from multiplicative number theory. The first
theorem can be found in [33, Section III Theorem 3.5]

Theorem 4.1.6. Let g be a non-negative multiplicative function , which, for suitable con-

stants A and B, satisfies
i.

Y_ 9 ( p ) logP Ay (y 0 )
p y

^

£ 9 (P “ ) logpQ sc B.

* Zp Z
=2
a

^
K

Then , for x > 1 we have

n x

° n x

^

^

28

The second theorem is a deep result from multiplicative number theory which

does much of the heavy lifting for the proof of Theorem 4.0.1. This theorem is due

to Hildebrand [9, Theorem 2]

,

Theorem 4.1.7. Let x and z be real numbers such that 2 y z y x be real numbers . Let g
be a multiplicative function supported on squarefree integers such that 0 y g ( p ] y K for

some K

Y_

1 and all primes p . Then zve have
r (K-i )

-

g ( TL )

-

*

r(K)
(1 - g (p ) ) +

P

)) (, + 0 (

( SH )’
))
(
)}
®

>

'

+0 e

where y is the Euler-Mascheroni constant , y + = max ( y , 0 ), (3 > 0 is an absolute con_
stant , ofu ) is a continuously differentiable function of u that satisfies <r ( u ) > u u and all

implicit constants depend only on K .

4.2 Proof of Theorem 4.0.1
Following the idea of previous authors (see, for example, [15, 23, 16]), we introduce
the following multiplicative function hx supported on squarefree integers.

Let a : [0, 1] -> [— 1 , 3] be defined by a ( 0 ) = 3 and <x ( t ) = 2 cos ( yyy ) + 1, if
yyy < t y; y where ra is a positive integer. Let N be a positive integer, and set hx
to be given by

hx ( p ) =

“(

iofiO if ^ x and f
P

otherwise.

0

29

p N

(4.2)

Lemma 4.2.1. Let f e IK£ ( N ) . Letx

^ l be such that Asym ( n ) Ofor all n < x. Then
2f

Y_ Asym ( n )

^^

2f

^

Proof . Since Asym2 f ( n )

hx ( n ) .

n x

n x
( n, N ) =1

^ 0 for all n y x, then
b

Y Asym fM Y Asym ( -r0,
2

2f

n x
( n, N ) =1

n x
( n,N ) =1

^

^

where Y.’ restricts the summation to squarefree integers. We will show that

Asym2 f ( n )
for all squarefree integers n

Hx ( n )

x with ( ri, N ) = 1 .

Since hx ( p ) > 0 for all primes p, and hx is multiplicative and supported on

squarefree integers, it suffices to show that Asym2 f ( p )
with p|hi . To this end, let p

hx ( p ) for all primes p

x

x be a prime number not dividing N , and let

m be an integer such that x ^' r < p y xm . Then for all 1 y j y m we have
xm + i < pi y x

0

y x, and so

Asymh (Pi ) =

sin ( ( j + 2 ) 8f ( p ) ) sin ( ( j + 1 ) 9 f ( p ) )
sin ( 20 f ( p ) ) sin ( 0 f ( p ) )
sin ( ( j + 2 ) ( 7T — 8f ( p ) ) ) sin ( ( j + 1 ) ( 7r 9 f ( p ) ) )
sin ( 2 ( TT - 0 f ( p ) ) ) sin [ n - 0 f ( p ) )

—

(4.3)
(4.4)

We used (2.14) to obtain the right hand side of (4.3).
Recall that 0 y 0 f ( p ) y n. Since sin ( 0 f ( p ) ) > 0, applying (4.3) with ) = 1 gives
sin ( 30 f ( p ) )

0. It follows that 0 y 0 f ( p ) y f or

^

y 0 f ( p ) y n.

. In this case sin ( 20 f ( p ) ) sin ( 0 f ( p ) )
First, suppose that 0 y 0f ( p ) y|

30

0. By (4.3),

we get

sin ( ( j + 2) 0f ( p ) ) sin ( ( j + 1 ) 0f ( p ) ) y 0,

(4.5)

for all 1 y j y m. We shall use induction to verify that 0 y 0 f ( p ) y
1 y j y m.

^

for all

The base case is already established since 0 y 0 f ( p ) y f . Now assume that
0 y 0f ( p ) y

for some 2 y j y m. By (4.5), we get sin ( ( j + 3 ) 0f ( p ) ) y 0 which

forces 0 y ( j + 3 ) 0f ( p ) y n, and so 0 y 0 f ( p ) y
Next, suppose that

^

as desired .

y 0 f ( p ) y n. In this case 0 y n — 0f ( p ) y|
. Repeating the

above argument with 0f ( p ) replaced by n

-

0f ( p ) and using (4.4), we establish that

for all 1 y j y m. Hence, if 0 y 0f ( p ) y j , then

0 y n — 0f ( p ) y

Af ( p ) = 2 cos ( 20 f ( p ) ) + 1 y 2 cos
Similarly, if

+ 1 = Mp ) -

y 0 f ( p ) y n, then

Asljm2 f ( p ) = 2 cos ( 2 ( 7t - 0f ( p ) ) ) + l y 2 cos

+ 1 = h* ( p ) -

We conclude that Asym 2 f ( p ) y hx ( p ) for all p y x with p \ N , and thus

Y Asymh (n) Y hx ( n ) -

n x
( n,N ) =1

n x
( n,N ) =1

^

^

a
In view of Lemma 4.2.1, proving Theorem 4.0.1 reduces to finding a lower

bound for

Y_ hx (n ) . To this end, we shall apply Theorem 4.1.7 with gfn ) =

ns£ x
( n,N ) = l

31

hx ( n ) and K = 3 which gives rise to the quantity

n (' + ) - n (

^

v

p

7

hx ( p )
1+ P
ps;x v
pfN

£: log ('l +

= exp

ps x
pN
\|

xexp

^
V

)
J

(4.6)
psjx

y

pN
\|

which follows from Taylor series expansion of log ( 1 + x ) and properties of hx.
In the following lemma, we compute an asymptotic estimate for the sum

appearing in (4.6) .

(

Lemma 4.2.2. Let e > 0 be given . For logx > max o) ( N ) 1 + e,

^

we have

,

M = 3 ogiogx _ 3 iog (J l_ ) + 0 ( 1 ) .

h2

pfN

Proof . We have

z ( (+ ) + ,) y I

^

2 cos

psix

PfN

x m 1I < p xm

PfN ^

= Il + I2,
where

Z- (v2 “s (v
’ KmsSM
=

^
32

) + 1)

7

7

H

—!—
X m I 1 <P

PfN

< xm

1
P'

x

and 1

-

X m+1

<p

|
pN

v'

^

xrri

is some parameter to be determined later. Observe that

M

=

1

L

L
Km <

L

V

M

' xm + <p
'

pfN

1 KmL
P

’
X i

^"

J-

+

M

_L

^ Xta+ <p^
1

1

~

i
xTn.

PtN

^

- In + 112 •

By Lemma 4.1.5, we get

111 = 2
1

=2
1

Y. cos (
L cos

^

2

-

) (‘ogiog ^ - iog ^s ^ ’ + 0 (

^

+ y 1))
|

m+1

tu ( N ) \ \

logx

Xm +T / /

Using the Taylor expansions for cos ( x ) and log ( l + x ) , we get
cos

(

^

Hence,

)

Cvr ) (' ( ) ) (i (i) ) i ( )

^
- (i (^) (

1 log

'

-

2

+0

I +0
« v

1

M

= 2 log M + 0

2

+0

m+1
+ 0 logx
/
\

\

M2

+ logx

-

+0

^

2

•

co ( N]\ \
Xm +! / /

Mcu ( N )
X JVl + I

Using Lemma 4.1.5 again gives
(

I12 = logM + 0 (

M

+
logx

By our assumption on x, we have logx

33

co ( N ) \
X M+T )

tu ( N ) .

Hence, we can choose M =

. This choice of M yields

h = hi + I12 = 3 logM + O ( 1 ) .
Next, we estimate I2 . We have
1

12
M< m

^

xm
N^
|
p

(’ + o ( ))

^
-( i
r
^ L

' '*»

z

1
+
p

p <x M + 1
pN
|

^

£ P

-

xS+T <p!$xffi

-

p x M +1
|
pN

|
pN

E

1

^§

L

M < TTl K I x m + 1 < p x
p
|N

\ m+ l <p

W2

L

+

-

£. f

p < x (VO 1
P|N

By Lemma 4.1.5, we have

Y
p <x M +1

PfN

^

= log log x M + I + CN + 0

^

M+1

ce ( N )

logx

X M+1

= log logx — log ( M + 1 ) + CN + 0

It follows that

^

M+1

logx

o> ( N )
XM + l )

I2 = 31oglogx - 31og ( M + 1 ) + 3CN

+0

M+1

logx

cu ( N )

log logx

x M+ i

M2

+

log ( M + 1 )
M2

= 3 lo§ log X - 3 log M - 3 log
+0 1+

(

tu ( N )

log logx

lo§ TW

XM + 1

M2

M2

34

)

| CN |
+ M2

1

All together, we now have

Ii + I2 = 3 log log x — 3 log

+

^

then logMlogx
2

Since logx > max

tu ( N ) 1 + e,

°

h + I2 = 31og logx - 31og

+

log log x
M2

M2

= 0 ( 1 ) . Therefore,

m2

( y) + 1

^

which completes the proof .

0( )

^

Lemma 4.2.3. Let e > 0 be given. For logx > max o> ( N ) 1 + e,

^

hx ( u )

Proof . To estimate
0

p x

m = 1 . Hence,

HX ( P ) ) + , notice that 1 —
p

( 1 - hx ( p ) ) + = -2 cos

Therefore, if z

N3

-

PtN

we have

4> ( N ) 3 x ( logx ) 2

+
(1

J

2 cos

hx ( p ) > 0

==> x 2 < psCx.

<

x 2 we have

jz<p x
PIN^

(1 — hx ( p ) ) +

V

L 2 = l o g 2 + o ( logx
1

=

X2

<p

P| N

^

cu ( N )
N

X

Applying Theorem 4.1.7 with z — 2 and K = 3, and Lemma 4.2.2, we get
e 2y e y
r (3) logx (
"

^^

n x

hx ( n )

x

los *>

3

(1r )

Piogxy

+ 0 [ e U°gA
35

3

f

, + 0 ( 5 ))
K uI

(2

^

)>

In obtaining the above inequality, we also used Merten's formula [24, Theorem 2.7(e)]
to estimate the factor EIpscx

( ^ _ ) appearing in Theorem 4.1.7. We conclude that
p

y

^
^’

n x

^

=

x ( logx ) 2

( mff

^)

^i

( ff ( 2 ) + o ( 1 ) )

4> ( IN ) 3 x ( logx ) 2

( 2) + 0 ( 1 )

N3

as claimed.

Finally, we are ready to prove Theorem 4.0.1.
Proof of Theorem 4.0.1 . By Lemma 4.2.1 and Lemma 4.2.3, we have

Y<-

Asym2 f ( TT.)
n <x
( n, N ) = 1

2

CNx ( logx ) ,

n x

where CM is a positive constant depending only on N . More precisely, we can take

_e

3T q ( 2 )

-

CN -

T(3)

/ 4> ( N ) \ 3
\ N y

"

4.3 Proof of Theorem 4.0.2
Recall that the symmetric square L-function associated with f is given by
L ( s, sym2 f ) =

Y£_ WfW
ns
n=1

1

'
(
)
r' no -T
PS

g — 2i 0 f ( p )

i

/

36

-

p

-

-

p |N

p

Taking the logarithm of (2.11) yields

logL ( s, sym2 f ) = -

^
^
pfN

2i0 f ( p )

(Vlog (V1 e V s

+ log 1 -

“ e 2 i Q 0 f ( p ) + e 2i a 0 f ( p ) + i
-

LZ

apQS

pfN a =1

=L
—

Vs

+ log ( l - P s )

logd - p-1-5 )

PIN

=

g — 2i0 f ( p )

_—
00

a
+ /y Zy
—
. Qpsa
p|N a =1

n-

P

A ( n ) bs m2 f ( n )
^ (n)
ns log

where

__

__

1 | g 2ia 0 f ( p ) | g — 2ia 0 f ( p )

bsym2f W = P

”

if u = pa , p \ N

if n = pa , p|N

Q

otherwise.

0

Let P ( n ) denote the largest prime divisor of n. For Re ( s ) > 1 , we set

Ly ( s, sym2 f ) =

ns

P ( nKy

n ('
PW

PtN

-

n
^
i

i
^9, ,Pl ' ( i - p r'
- p * ’)
nO
ps

e
-

)

-

-

- -

p <y

PIN

(4.7)

We refer to Ly ( s, sym2 f ) by the partial symmetric square 1-function. We have

logLy ( s, sym2 f ) =

Y=_

ns log u

(4.8)

logics, sym2 f ) — log Ly ( s, sym2 f ) .

(4.9)

n 1

P ( nKy

Next, we aim to estimate the difference

37

To this end, we require the following lemma .
Lemma 4.3.1. Lef f G Jf £ ( N ) . Lef s = cr + it with

\ < a sC

Let

suppose that L ( z, syrrrf ) has no zeros within the rectangle { z : <T0

\ < a0 < a, fl /zd

Re ( z )

1 , Im ( z ) -

t| y 3}. Then we have

log L ( s, sym2 f ) <

^

log N 2 (Vk2 + t 2 + 3
CT - ff 0

^^

(4.10)

Proof . Consider the circles Cr and CR centered at 2 + it with radii r = 2 - a and
R = 2 — Do respectively. Observe that s G Cr . Since L ( z, sym2 f ) has no zeros inside

CR by assumption, then logL ( z, sym2f ) is analytic there. For z G CR , the convexity
bound for symmetric square L-functions (2.15) gives

Re ( log ( L ( z, sym2 f ) ) ) = log | L ( z, sym2 f ) | < log ( N 2 ( |z + k| + 3 ) 3 ) .

Applying Theorem 4.1.1, we find that

I

max | logL ( z, sym2 f ) |

max Relog ( L ( z, sym2 f ) )

+

lo§ L 2 + lt sy -2 f ) l '

^

For z G CR, we have \ z + k| < \/k2 + 12 since Im ( z ) < 2 + 1 — OQ < t . Hence,
max

^

Relog ( z, sym2 f )

^ ^ ^^

log N 2

Returning to (4.11), we obtain

|l°gL ( s, sym f ) |

/

\ k2 + t2 + 3

2

a - CT0

38

(4.11)

A crucial step in estimating (4.9 ) is estimating the two following differences
A ( n ) bsymAN
ris log n

log L ( s, sym.2 f ) —

(4.12)

and

logLy ( s, sym2 f ) -

^

A ( n ) bsymAW

(4.13)

nslogn

In the following two lemmas, we find asymptotic upper bounds for (4.12) and
(4.13) .

Lemma 4.3.2. Let f e 5f £ ( N ) and assume the GRH for L ( s, sym2 f ) . Let y

^ 2 and

1 . We have

s = cr + it with a

2 )

logL ( s, sym f =

^

nSl § n

^

^ log N ( V^TIRT

°

<

n y

2

^

+ 3)

^

Proof . Without loss of generality, assume that y 6 Z + ], Let ci = 1 — a +

( logy ) 2

^

(4.14)

. By

Theorem 4.1.2, we have

AMbsym2f M
1
logL ( s + z, sym2 f ) — dz =
2m Jcl us log n
Z
u y

f

^

^

°

i:
< <

“

„K,

where
R=2

n

^ / 4 + yCl
<n < y

+o

Cl

V

y

AMbsymAM
si M lo8
ns
TL
log
«
y n 2y
I A ( rL ) bsym2 f ( Tl- ) I 1

I

nS

losn

InCl

39

(

( ))

Since si ( x ) satisfies si ( x ) < min (1, b ) , we have

R«

A N | bsym2 f W |
+0
(s)
nRe
logn
log
(
)
|
[
<2 y

—y y

^

^

|
( t uRe| bsymh
logn
A ( n)

V

-

(n)

(s)

1
nci

.

(4.15)

Investigating the first term in (4.15), we see

y

fi^
«

V

AWlbsym2 f ( ^ ) l
( l0g U ) ll0g ( ) l
2ynff

1

L

~

y CT+T

1
« y CT+T

-

4 < n < 2y

^

A ( n ) lbsymh ( n ) l
( logn ) |log ( ) |

^

1 + gi2a0 f ( p ) + e-i2a0 f ( p )

L

1
yff +1

a M u)|

< pa < 2y
pfN

p~ °

^ ^

y<

— ^ Kwr

«y

1

1

4 < pa <2y

Let r = pa — y . Using the inequality log ( x + 1 )
estimate

l

ft . Hence,

MA )
1

1

^^

^h

liog ) !

1

<< y"

^

2 <r <

r =0

^
1

1

we get

y | AWbsymhW | J _ « r"I 1 1

^ I
“

T

-

nc

nCTlogn

“

0< r < y

As for the second term in (4.15), using c -\ = 1 - a +
y
y

— 1 , we obtain the

L ; « « los* -

3

u

PIN

for x >

1

| °g ( ) |

2y a l

n =1 n isk

+

< y Hogy ,

showing that
1
2ni

[

A

logL ( s + z, sqm2f ) — dz =
Z

nsCy

40

+ (y Hogy ) . (4.16)
^nsbsymA
log
)

(n)

n

0

-

\

Observe that when C2 < Re ( z )

Letc2 = —

ci , we have Re ( s + z ) e

By taking UQ = \ in Lemma 4.3.1, we get
log L ( s + z, sym2 f ) <

log ( N 2 ( \/ k2 + ( |t| + Im ( z) ) 2 + 3) 3 )

\

Re ( s + z ) -

log ( N 2 (

Vk + ( |t| + rj ) 2 + 3) 3 )
2

1

logy

^^ ^

< log N 2

k2 + ( |t | + tJ ) 2 + 3

(4.17)

iogy

uniformly for z, with Re ( z ) L C2 and | Im ( z ) | L y .

-

We now move the contour of integration in (4.16) to the line Re ( z ) = cz By our

assumption of the GRH, the integrand only has a simple pole at z = 0. The residue
theorem then yields the following
1
logL ( s + z, sym2 f ) — dz = log L ( s, sym2 f ) + £,
2m Jci -iy
z

[

(4.18)

where

£=

1

/

rc2 iy

fC 2 + iy

-

,

rci + iy

L LB

+

ts

-

z

) logL (s + z' sym f Tdz

+

'

Using (4.17), we observe that

J

logL ( s + z, sym2 f )

^

- dz < log

Cy

1

"

<

^^^^ ^
^^ ^^ y^^
N2

y k 2 + ( |t | + y ) 2 + 3 ^

j logyj yj- da

^

logy log N 2

< y 1 +Cl log

N2

_<

(

”

Tlog N 2

41

/k + (|t| + y ) + 3) |
j yQda
2

2

j

k2 + ( |t| + y ) 2 + 3

k 2 + ( |t | + y ) 2 + 3

.

The third integral,

[ logL ( s + z, sym f ) —z is treated similarly. For the second
Jc +
2

2

iy

integral, we have

J

^

| |logL ( s + c + , sym ) | j

logL ( s + z, sym2 f ) - dz <

2

ib

2

^

f

< yC 2 logy log

^
^
^ ^ - ^^

< yC log
2

(

N

2

)

lc2

y

_1 . [

27X1 JCl

3

+ (|t| + y ) 2 + 3

log N 2 ( \/k2 + ( |t + y ) 2 + 3

Hence,

^jdb

r

-

( logy ) 2

( logy ) 2

i

logL ( s + z, sym2f ) — dz = logL ( s, sym2 f )
Z

^

^ log N ( v k + ( |t | + y ) + 3)

+0

2

/ 2

2

yff — 2

^

( logy ) 2

as desired .

Lemma 4.3.3. Let f

dt£ ( N ) . Let y

2 And s = cr + it with a

logL# ( S, sym2f ) -

_J

«

1 . T/icn

1

Vv

'

Proof . We have
T
(r s, sym2fxi) - V
ll o g Ly
n< y

£

A ( U ) bsym2 f ( n )

^

b

=

A ( n ) bsym2 f ( n )
ns log n
p «Jy
pa > y

L

Z> Z
a 2

42

v>y «

^

«Z

a>2

(d )

^

— CKT+1

« y -° + 2 .
'

1

^

Since cr > 1 , we get the desired result.

In the following proposition, we return to the task of estimating (4.9 ) .
1H£ ( N ), and assume the GRHfor L ( s, sym2 f ) . Let y

Proposition 4.3.4. Let f
s = cr + it zuith cr

2 and

^ 1 . Then

logL ( s, sym2 f ) - logLy ( s, sym2 f ) <

^

Vk + ( |t| + y ) + 3)

( logy ) 2 log N 2 (

2

2

v/y

Proof . By Lemma 4.3.2, Lemma 4.3.3, and the triangle inequality, we have

^

|log L ( s, sym f ) — log Ly (s, sym f ) |
2

2

< logL ( s, sym2f ) —

ANbsymbN
+ logLy ( s, sym2f ) ns
n
log
n y

^

<

Vk + (|t| + y ) + 3)

( logy ) 2 ( log N 2 (

Since a

2

2

-

y° 2

1 , we obtain the desired bound .

^

^

A ( u ) bsym*f ( n )
ns log n

1

Vv

'

In the following lemma, we provide an asymptotic upper bound for Ly ( s, sy m2 f ) .

Lemma 4.3.5. Let 2 < y

^

x, and let s e C be such that Re ( s ) = 1 + j

Ly ( s, sym2 f ) < ( logy ) 3.

. Then

, zve have

Proof . Using (4.8), we write
log Ly ( s, sym2 f ) =

Y_

21 e2ai0 f ( p ) + g — 2ai0 f ( p ) + |-

e2i0 f ( p ) + e-2i 0 f ( p ) + i

Ps

p y

^
PfN

p y a =2

^

pfN

°° -n a
-

p ya 1

^
PIN

43

aPas
(4.19)

Investigating the sums in (4.19) individually and using ( 2.9 ), we see that
e2i0 f ( p ) + g —2i0 f ( p ) + |-

Vs

p sy

=

p sy

=

PIN

=

^

Vs

pfN

2. g2ai0 f ( p ) _|_ g —2ai0 f ( p ) + |-

apas

P Sy a =2
p|N

-L

Af ( p2 )

__

3

< 2V 2V apaRe(s ) = 0 ( 1 ) ,
p y a =2
pfN

^

and similarly

°°

0( ).
F aF —
apas = 1

psSy
p |N

= li

Now we can write

Ly ( s, sym2 f ) = exp

MEJ + 0 ( 1 ) ,

^

P

p <y

\ p{N

which shows

<<

X_

^
^
expfe) ^

Ly ( s, sym2 f ) « exp

j

Af

< exP

3

j

< exp ( 3 log logy ) = ( logy ) 3.

Now we are ready to prove the following important result from which Theorem
4.0.2 is deduced . In this result, we show that the partial sum

Asym 2 f ( n ) can be

approximated by the corresponding sum of Asym2 f ( n ) when restricted to integers n
such that P ( u )

y , when assuming the GRH.

44

Proposition 4.3.6. Let f e

( N ) , and assume the GRH for L ( s, sym2 f ) . Then for all

^

real numbers x, y such that ( logkN ) 2 ( loglogkN ) 10

Y<

Y
nsgx

Asym2f N =

u x

Asymh M +

P(nKy

°(

y <x

kN we have

x ( logx ) ( logy ) 5 ( log ( kN ) )

x

Proof To simplify the exposition, we set

Ssym2fW = HAsym2fW

Asym fM ^ (^ y; Asym^ ) = Y
rWx

and

2

f

n x

^

PfnKy

Without loss of generality, we assume that xeZ + j. Let ci = 1 +

By Theorem

4.1.2 we have

xs
1 rci +ix
Ssym fW = f f f i _ . L ( s, sym2f )-ds + R,
c
x

\

2

where
R=

iY

^ ^

AsymhWsi (x l o g ( )) -

2 <n < x

4Cl + XC 1

+0

X

^

Y AsymhWsi (xlog ( ))

x < n < 2x

y | Asym fM | \
2

^—

nci

n =1

J

By the triangle inequality, we have

R<

|Asym ( n ) | + 0 4 + Xci |Asym (n ) |
^= nci
tf 2xxllosk ) l
«m ,(n) |
(
K
f
| ( )
2xxl os s l
}
\h n"

1
7

^
^

2f

^—

ci

X

2f

n 1
2

<

|

|

To estimate R, we search for an upper bound for Asymzf ( TL ) . We use (2.13) which

45

|

gives the Deligne-type bound Asymif ( u )

y

R<

-

X

X

2 < n < 2x

X

|

T3 ( U ) . Using this estimate, we get

As A)

lloS ( t ) l

n x

„

^ (n)

y T3
~

n =1 n

( -n )
- u x |!og ( 5 )
< n < 2x

L
f

4^

1+i

r

(

(4.20)

For any e > 0, the

Observe that the first sum in (4.20) is equal to C l +
second sum in (4.20) is bounded as follows:
T3 ( U )

y

1

n2

V

|
< n < 2x

|iog ( hi

1

1_
X 2

1
2 < n < 2x

ll0S ( ^ ) l

L

_
yA
x1 -! x |r|
M<

< x -2 logx C xe,
<

where we set r = n

— x, and the implied constant depends only on e. This yields

SsymbM =

1

fC 1 +ix

^J

.

-

yS

L ( s, sym2 f ) yds + Oe ( xe ) .

Applying the argument we used to establish (4.21), we get

^

, YS
i rci + ix
( x/ y; ASym 2 f ) = —
Ly ( s, sym2f ) yds + Oe (

J

xe ) .

To analyze the difference of Ssym2 f ( x ) and Tfx /y ; Asym2 f ) , we define

Ry ( s, sym2 f ) =

46

L ( s, sym2 f )
Ly ( s, sym2 f ) ’

(4.21)

We have

SsynWM - ^ (*, y; Asym2 f )

( L ( s, sym2f )

=

= 27h

I

sym2 f

Ly

-

Ly ( s, sym2f )

) y ds + Oe ( xe )

^ (exP (log ^ sym f 0 1 )
2

S/

“

~

ds

+ Oe ( xe ) .

(4.22)

By Proposition 4.3.4, we know that
(

^

( logy ) 2 log N 2

2 )

log Ry ( s, sym f = 0

(yk2 + (|t| + y ) 2 + 3)
Vy

^

Observe that for kN large enough, we have

^

( logy ) 2 log N 2

since kN

(

^
^
log N 2

Moreover, since 10
|

\ y

/

Vy

^

= o( l )

( log kN ) 2 ( log logkN ) 10 . To see this, first notice that

x>y

( logy ) 2

(Vk2 + ( |t| + y ) 2 + 3)

^
"

J

/ 2

2

y

V k + ( |t | + y ) + 3

^

C log kN .

<

is a decreasing function for y large enough, we have

(log ( ( logkN ) 2 ( loglogkN ) 10 ) ) 2

J

\

( log kN ) 2 ( log log kN ) 10

log log kN
^ logkN ( log logkN ) 5

^

^

^

It follows that
( logy ) 2 log N 2

( k + ( |t| + y ) + 3)

^

2

2

Vy
47

1

< ( log log kN ) 4 '

'

which tends to 0 as kN -+ oo . Hence, we can apply the approximation ex = 1 + O ( x ),
which is valid for 0 < x < 1 , to get
1

( (Vk + (|t| + y ) + 3)

( logy ) 2 log N 2

exp ( log Ry ( s, syrrrf ) ) = 1 + 0

2

2

Vy

for all s with Re ( s ) = ci and |Im ( s ) |

^

(4.23)

x. Applying Lemma 4.3.5 and (4.23) in (4.22)

gives

Ssym+ M -

^^

y stpuA ) <

<

( logy

) 5xCl

Vy

—

log ( u 2 (yh2 + (|t| + y ) 2 + 3

:i
X

X

xlogx ( logy )5 log ( Nk )

7

'

) j
“

dt

as desired .

The proof of Theorem 4.0.2 will follow from Proposition 4.3.6 and the following
lemma, wherein we bound T ( x, y; T3 ) .

Lemma 4.3.7. Let 10 + y < x be real numbers . Then zve have
¥ ( x, -y; T3 ) < e

as u

2X

( logx ) 2

_ iQRx —> oo.
•

logy

Proof . Recall that
¥ ( x, y; T3 ) =

Y_

T3 ( n ) .

n x
P (nKy

^

Let p = 25 k

^

< Y ( logY ) 2 (see [24, page 43]), we

- Since T3 ( n ) > 0 and

48

have

Y

¥ (*,!; T3 )

*

T3 ( n )

3

x 32 < n < x
P ( nKy

TWX 32

Y

Y nl n

+

T3 ( TX )

-

+X

n x 3l

^ ^ ^ tn
n

X 32 < n

^

)

^x

P ( nKy

^ ^^

< x 3§ ( logx ) 2 + x 3

n|3 T3 ( n )

U X

p (nKy

< x 32 ( logx ) 2 + e ¥
n x
P ( n ) Cy

^

We now proceed to estimate

^^

n T n).

~

(4.24)

Y< ^ ^ ). Let
=
P

T

U

n x
P ( rL ) < y

9 (n) =

n|3 T3 ( n )

i f P ( n)

0

otherwise.

^y

Since T3 is multiplicative, it follows that g is multiplicative as well. Hence, for all

primes p

y the following statements are true:

g ( pa ) = v |3
q

( a + 1 ) ( a + 2)
2

and

£

>* < ( )
Next, we prepare to apply Theorem 4.1.6 by verifying the following inequalities:
i.

^

g ( p ) logp y Ax

P

a =2

ps;x

P

49

for some positive constants A and B . Observe that

Y_ g ( p ) logp = Y_ 3PP logP 3e ^ Y=sy- loSP 3e “ V
p x

p y

^

/

p

^

^

where the last asymptotic holds by the Prime Number Theorem in the form Ap y logp ~

y . This gives (i) since y < x. Next, we have

,

£2 ( a + l ) ( a + 2 ) pQ ^ logpa
V

2Pa

pCy a=2

a=2

=

P y

Z

2

a 2

^

=

22 a ( a + 1 ) ( a + 2 )

Y OSVY=
1

_

1OSPP2 ( p 1

P y

T=
a 2

^

=

22 a ( a + 1 ) ( a + 2 )_ ( 1 ) ( a 2 )
—
p P—
2

YloSPP

2( H I

p y

^

£ +
(a

2 ) ( a + 3 K <n + 4 )

a —0

The inner sum converges to some constant since pp 1 y

^

for any p

( 3 _ 11 ( a

^

y . Next we

consider the outer sum

r- loSn
XlogPP^ U Z
- U2 ( 1 P )

^

p y

^

which converges if |3 <

-

n =1

^

Since (3 < 25 ] g 4 < 0.47, we obtain (ii). Observing that

Y_ rtpT ( n) = Y_ gW
3

n x
P( n) y

n x

^^

^

50

and applying Theorem 4.1.6, we get
x

Y_ npT ( n )
3

y

n

n x
P ( nKp

^

n^ (' + ^ ’ i ’
= i n (, pM ) n (
^

^
^ no ^on
^
=i 6

- +

p p V

a=2

( o + i ( o + 2)
2pQ
i+

+3

°

<

p y

p p

1+

+

^

p=i

P P

( a + 1 ) ( a + 2 ) pa|3

1

2Pa

\

OO

&

pae

^

1
1 + 3pP i q
-

=2

( a + 1 ) ( q + 2 ) pa
2PQ

^

^

It is well-known that an infinite product ] [ ( 1 + an ) with an > 0 is convergent if
n=1
oo
and only if the infinite sum

^

an is convergent. Since

n =1

^

- ( a + 1 ) ( a + 2)
1
- p2 tp-h y- ( a + 3 ) ( a + 4 )
( P -1 )
y
pa
pafP-1 )
_1
=
1
—
pp
1 +3
2
1
2
3
H r
*a 2
*a 0
p

^

—

J

—=

«

—=

,

p 2{ (3 — 1 )

1

ZTJhP^ ^ n
P

K

n

n2

it follows that

n h + + PP '
OO

1

1

P =1

3

“

Therefore, we have

^

( a + 1 ) ( a + 2 ) pnP
2 p“
,

0

2

Y<_ npT ( n)
3

n x

PlnKu

51

< oo.

-

P) <

,

OO

Applying this estimate in (4.24) yields

^
Observe that xM

(*, y; T3 ) < x 3i ( iogx ) 2 +

xe 21osy = xe

2

for y

xe

~

^ n (v > + f )

logx

J

P +y

10. Moreover, we have

4+'+))

n (' ++»
Notice that

since x < 1 + 2 logx for 1 + x + 3. It follows that

n^ ( v ) sexpf\ ^ +
p y

v

1

i+

K 7

p y

0 ( Plogp )

P

4£ M'^))

"

= exp ( 3 log logy + 0 ( 1 ) ) < ( logy ) 3.
Thus,

4/ ( x, y; T3 ) < e

2 X(

logx ) 2 + e 2 —

logx

(l o g y )3 < e

2

x ( logx ) 2,

as desired .

Finally, we are in position to prove Theorem 4.0.2.
Proof of Theorem 4.0.2 . Suppose x + kN and choose y = ( logkN ) 3. By Proposition

52

4.3.6 we have

Y- Asym ( n ) =

Y

2f

Asijm 2 f ( n ) + 0

n x

n x

^ )
PfuKflogkN

^

3

^
^

3
C V ( x, ( logkN ) ; T3 ) + 0

<

< x ( logx ) 2 exp
= o ( x ( logx ) 2 ) .

(V

x ( logx ) ( log ( logkN ) ) 5 ( log ( kN )

x ( logx ) ( log ( logkN ) ) 5
( logkN )!

log *
2 log ( logkN )
-

( log kN ) 2

( log ( log kN ) ) 5 xlogx

+0

( log kN ) l

Now all that remains is to verify this asymptotic for x > kN . To do this we estimate

\

the integral appearing in (4.21) . Via shifting the line of integration to Re ( s ) = + e
we see
fCl +ix

Jci — ix

s

L ( s, sym2 f ) — ds =
s

^
^

/ r + e +ix

r ci — ix

—

yJ +e — ix J 4 + e-ix

f 7 + e + ix \

J c! +ix

)

s

L ( s, sym2f )- ds = h - I2 - I3.
s

For 2 < Re ( s ) < 1, the sub-convexity bound for symmetric square L-functions
(2.2.1) shows

log ( lSI 2 k2 ( |s + 3|3 ) ) 2 2 Re ( s ) \
loglog ( N 2 k2 ( |s| + 3 ) 3 ) )
~

Us, sym2 f j c exp
<

53

'

Using this estimate, we get

^
^
^£ (7G

log ( Nk )

Ii < x 2 + e exp 2 C loglog ( Nk )

[ exp 3C
Jo

< x + eexp

° ( v ( i + e ) + t + 3)
/

1g

l glog

°

2

2

^

1

( V ( + ) + + ) J ]( + ) +
e

l

2

t2

3

\

e

2

(

2

^

t2

)

l g
i + eJ + t
°
1
log ( NITc) \ / r t o
v
- AT
2C
exp 3C = dt
2
2
loglog ( Nk ) 3 V
3
e
t
/
loglog V ( + e ) + 3 j V ( 2 + ) + 2
°

+

2

,

2

2

2

^ ^

) + t + 3) \/ ( + 1e ) + »
2

+e

2

j

dt

2

For t 0 large enough, the first integral is a constant depending on t0 ( which in turn
log
depends on e ), and the second integral is at most xe . Hence h C xi + e exp (' lC loglog
( Nk ) )
( Nk )

<

We also get

} jL (u - ix, s,m^ f )| l= du
2

^

du = Ji + J 2 .

We have
i
3C log (Vu + x + 3
log ( Nk )
Ji < xexp 2 C log log ( Nk ) ) Ji exp
+e
log log Vu2 + x2 + 3

^
^

2

•

(

log ( Nk )

< xe exp 2 C loglog ( Nk )
since v7u2 + x2 x x. In the range 1

cr

2

^^

1

, the second term in Theorem 2.2.1 is
|

54

\

dominating, and so we see that

.

(log ( N k ( |u — ix| + 3) )) V U 1+ dU
xCl'
og ( NV (, + 5 3)) %<
xlogx (
^xp (ci« ),

J 2 < xCl

JCl

2 2

3

,

A

2

X2

.

for some absolute positive constants A and C. The integral 13 can be estimated

similarly. Hence, we have
log ( Nk ) \

(n ) < x + eexp fa
lAv
loglog ( Nk ) y '
rWx
2f

2

^

for some absolute positive constant C -\ . In view of this asymptotic upper bound,
we see that under the GRH we have

^^

u

(

)

^
x

Sym 2 f

— o ( x ( logx ) 2 ) in the range x >

exp c2 logiogkisi l()r some absolute positive constant C2. In particular, the assertion of Theorem 4.0.2 holds for x > kN .

55

Chapter 5
Sums of Fourier Coefficients of

Hilbert Modular Forms
The purpose of this chapter is to extend [16, Theorem 1.3] and [16, Corollary 1.2]

to the setting of Hilbert modular forms.

This chapter is organized as follows. In Section 5.1, we introduce the theory of
Hilbert modular forms after which we state our main theorem for this Chapter. In
Section 5.2, we prove our main results.

5.1 Definition and Basic Properties of Hilbert Modu-

lar Forms
In what follows, we present standard material for Hilbert modular forms. The
reader is referred to, for instance, Garrett [6, Chapter 1, 2], Raghuram-Tanabe [25,
Section 4], and Shimura [32, Section 2] for more details. All the theorems discussed
in this section can be found in the aforementioned references.

Throughout this chapter, we take the base field to be a totally real number field
F of degree n over Q, and we denote its ring of integers by 0 F . The unit group in 0 F

56

is denoted by Op . The absolute norm of an ideal a in Op is defined asN ( a ) = [Op : a] .

In fact, the absolute norm defined as such can be extended by multiplicativity to
the group, 1( F ), of fractional ideals of F. The trace and the norm of an element x in
the field extension F/Q are denoted by Tr ( x ) and N ( x ), respectively. Notice that for
a principal ideal ( x ) = xOp, we have N ( ( x ) ) = | N ( x ) | . The different ideal of F and
its discriminant over Q are denoted by Dp and dp, respectively. We also have the

identity N ( 2) p ) = | dp|.
The real embeddings of F are denoted by Uj : x

Xj := cxj ( x ) for j = 1 , . . . , n.

Once and for all, we fix an order of the embeddings, say u

( crp , . . . , crn ) , so that

any element x in F can be identified with the n-tuple ( xp , ... , xn ) in Rn . This tuple
may be, again, denoted by x when no confusion arises. We say x is totally positive
and write x » 0 if Xj > 0 for all j, and for any subset X c F, we put X + = {x 6 X : x »
0}. Similarly, any element y

GL2 ( F ) can be viewed as an element y = ( yp , .. . , yn )

in GL? ( IR ) n via these embeddings.

We denote the narrow class group of F by Cl+ ( F ) . This is defined as the quotient

group I ( F ) / P+ ( F ) , where P+ ( F ) is the group of principal ideals generated by totally
positive elements in F. It is well-known that Cl+ ( F ) is a finite group, and its car-

dinality is denoted by hp . Henceforth, we shall assume that hp =1. We make this
assumption to avoid working with adelic Hilbert modular forms, the definition of
which is beyond the scope of this thesis. We also assume that the Dedekind zeta

function of F Cp ( s ) does not have an exceptional zero. This assumption allows us
to avoid technical issues when applying prime ideal estimates.

Let k G 2 Nn, and let 9F be an ideal in Op . We say f is a Hilbert cusp form of

weight k and level Tt if it is a holomorphic function on Hn satisfying the transformation property, f|T = f , for all y in F0 ( OF, Dp 1 ), where |Y is the slash operator given

by
f|y ( z ) = ( dety )

^

57

( y, z ) -kf ( yz ),

and
a b

r0 ( OT, 2) p 1 ) =

G GLj ( F ) : cydG 0 F / b G

c d

SJT1, c G 9TDF ad
/

-

bc G O x + V .

A Hilbert cusp form f admits the Fourier series expansion

^

f ( z) =

a.f ( v ) exp ( 2mTr ( vz ) ) .

(5.1)

xe ( oF ) +

We set Sk ( At ) to be the space of all Hilbert cusp forms f of weight k and level At .
The space of cusp forms can be decomposed as Sk ( At ) = S£ld ( At ) © S|)ew ( 0t ) where
S°ld ( At ) is the subspace of cusp forms that come from lower levels. The new space
S£ew ( 9t ) is the orthogonal complement of S°ld ( At ) with respect to the Petersson

inner product

(f 9 ) ot =
/

[ Jrodn
[ DpbvH f (z) g (z) ynkdji( z),
• •

J

(5.2)

^

/

,

,

where dy ( z ) = ] [ j = i yj 2 dxjdyj with z - = Xj + iy .
~~

The space

”

has an action of Hecke operators {Tm}mc 0 F much like the clas-

sical setting over Q. A Hilbert cusp form f in S fOT ) is said to be primitive if it is a

^

normalized common Hecke eigenfunction in the new space. We denote by nk ( 9t )

the (finite ) set of all primitive forms of weight k and level Al.

For an integral ideal m, there exists a unique (up to multiplication by a totally
positive unit) v G 0 F such that m = vOF . This follows from the assumption that F
has narrow class number 1. Given f G FTk ( 9T ) , we set

Af ( m ) = Of ( v ) ( N ( t n ) ) i ©

(5.3)

where af ( v ) is the v-th Fourier coefficient in (5.1). It follows from the work of
Shimura [32] that the coefficients Af ( m ) are equal to the Hecke eigenvalues for Tm

58

for all m. Moreover, Af is multiplicative in m, and the coefficients Cf ( m ) are real for

all m . Thanks to the work of Blasius [1], which proves the Ramanujan Conjecture

for Hilbert modular forms, we have the bound

(5.4)

Tf ( m ),

IAf ( m ) |

where TF ( m ) is the number of ideals in 0 F dividing the ideal m. This bound is

analogous to Deligne's bound for classical modular forms. We mention here that
much like the integer divisor function T, the ideal divisor function TF satisfies [35,

Page 194] TF ( TV ) < e N ( n ) e for any e > 0, and rF ( n ) is the coefficient of N ( n ) s in the
~

series representation of CF ( s ) . We recall that the function CFfsj is the Dedekind zeta

function of F.
Next we define the L-function associated with f e FTk ( 9T ) . We set

Y_ NAf( m )

(m)

L ( s, f ) =

s’

mcOp

Observe that L ( s, f ) is absolutely convergent for Refs ) > 1, and it admits an analytic
continuation to the entire complex plane. By multiplicativity of Cf , we have the

Euler product decomposition:
1

(1 - (3f ( p ) N ( p-s ) ) 1

p\0 X

n; (

1 - cxf ( p ) N ( p-s ) )

\ (5.5)

p|9t

where <xf ( p ) and (3f ( p ) are referred to as the local parameters of f and satisfy (see

[35, Proposition 4.2])

| of ( p ) l = Of ( p ) Pf ( p ) = 1
cxf ( p ) = ±

= , (3f ( p ) = 0

^ =

59

if p f 94
if p|91.

(5.6)

If p|91, we write af ( p ) = el6f ( p ) and |3f ( p ) = e l0 f ( p ) for 0f ( p ) G [0, 7t] . We also have

the relations

Af ( pv ) =

«f ( pr +1 -|3f ( pr +1
Of ( p ) - 3f ( p )

sin ( ( v + 1 ) 0f ( p ) )
, (v
sin ( 0f ( p ) )

^ l ).

(5.7)

The Generalized Riemann Hypothesis (GRH) for L ( s, f ) is the conjecture that the
non-trivial zeros of L ( s, f ) have Re ( s ) =

A convexity bound for L ( s, f ) is given by

(

| L ( s, f ) | « N ( 9l ) ( |s + k| + 3 )

+e

lnY .

(5.8)

The reader is referred to [13, Equation 5.22] for a detailed exposition on convexity
bounds for automorphic L-functions. Assuming the GRH, we have the follow-

ing much stronger bound which follows from [13, Theorem 5.19] when applied to
L ( s, sq m2 f ) .

. Assume the GRH for log ( L ( s, sqm2 f ) ) .
Theorem 5.1.1. Let s — a + it with \ < a |
Then
2 2a

log ( L ( s, sym2 f ) ) <

2 (log ( N ( 9t ) k2n ( |s | + 3 ) 2n ) )
+ 3 log log N ( 91) k2n ( |s| + 3 ) 2n
( 2 a — 1 ) loglog ( N ( 01) lc2n ( | s| + 3 ) 2n )

(

with absolute implied constants .

Having introduced the necessary notation and terminology, we can now state
the main theorem of this Chapter.

Theorem 5.1.2. Let f G nk ( 9t ), and assume the GRH for L ( s, f ) . In the range
oo, we have

Y_ Af ( n ) = o ( xlogx ) .

N (n) x

^

60

)

5.2

Proof of Theorem 5.1.2

Taking the logarithm of (5.5) yields
logl ( S f ) = —
/

£(

log

^

+ log

.

“

+ log 0 - N [
'

pn

)

logn - Ni ) -1-5 )

*

pm

22. eioL0 f ( n ) _|_ e-ia0f ( p )
+
oN ( p ) “

pfOt a=1

=

(' w)
£ N ( P ) -as

ZZ
aN ( p )
t =

Q

.

p|9 a l

AF ( n ) bf ( n )

L N ( n ) glog ( N ( n ) ) '

ncOp

where

AF ( n ) =

log ( N [ p ) ) if n = p a for some prime ideal £
otherwise,

0

and

eiQ0f(p) + g — ia 0f(p)

bf ( n ) = Nlp ) a

if N ( n ) = N ( p ) a, p ( 91

p \m

if N ( n ) = N

~

otherwise.

0

For an ideal n, we write P ( n ) y y if all the prime ideal divisors of n have norm
less than or equal to y . Consider the partial sum

logLy ( s, f ) =

bf
Y_ N AF
( n ) logN ( n )
(n)
s

ncOp
P(n ) y

^

61

(n )

*

(5.9)

Our first goal is to analyze the difference

log L ( s, f ) - logLy ( s, f ) .

(5.10)

To this end, we require the following lemma .
Lemma 5.2.1. Let f

f Let j

TTidOt ) . Let s = cr + it ivith \ < a0

suppose L ( z, f ) has no zeros within the rectangle { z : <T0

Re ( z )

o0 < a, and

l , | Im ( z )

— 1| < 3}.

Then we have

^

log N ( 9l )
log L ( S f )
/

( Vl? + t + 3)
2

a - a0

^
'

(5.11)

Proof . Consider the circles Cr and CR centered at 2 + it with radii r = 2 — o and
R = 2 - cr0 respectively. Observe that s e Cr . Since L ( z, f ) has no zeros inside CR

/

the function log L ( z, f ) is analytic there. The convexity bound (5.8) gives

Re ( log ( L ( z, f ) ) ) = log |L ( z, f ) | « log ( N ( 91) ( |z + k| + 3 ) 2n )

for any z G CR . Applying Theorem 4.1.1 we find that
2r
max_ JlogL ( z, f ) C max
— T z\

lz-

=

| logL ( 2 + it f ) |. (5.12)
+
RRelog ( L (z, f ) )|

^

/

For z G CR, we have |z + k| <C \/k2 + 12 since Im ( z ) < 2 + 1 — <J0 < t . Hence,

^

Vk2 + t2 + 3)

max Relog ( z, f ) < log N ( 9t ) (

\ z-

^

Appealing to (5.12), we obtain

^

log N (9l )

I log L ( s, f ) | <

( A + t + 3)
2

a - a0

62

2

^

2

^

By the triangle inequality, it suffices to find suitable upper bounds for

bf
Y_ N AF
( n ) logN ( n )

(5.13)

bf
Y_< N AF
( n ) logN ( n )

(5.14)

(n)

logL ( s, f ) —

(n )

s

N (nKy

and
(n)

logLy ( s, f ) -

(n )

s

N (n ) y

in order to bound (5.10) . In the following two lemmas, we find asymptotic upper

bounds for (5.13) and (5.14) .
Lemma 5.2.2. Let f e 71 ( 91) and assume the GRHfor L ( s, f ) . Let y

^

with cr

2 and s = a + it

1 . We have

log L ( s, f ) =

AF ( n ) bf ( n )
. N ( n ) s logN ( n ) + 0

Y

'log (

N ( 9t )

( Vb + (|t| + y ) + 3)
2

2

N (n K y

Proof . Without loss of generality, assume y e Z +

Let ci = 1 — a +

Theorem 4.1.2, we have
1

pi +bJ

2m Jci -

^

TJZ

log L ( s + z, f ) — dz =
Z

bf
R,
YK_ y N AF
( n ) logN ( n )
(n)
s

N («

63

^

(n )

( logy ) 2

(5.15)
. By

where

R=-

71

z—

AF ( n ) bf ( n )

Z
N ( n ) s logN ( n )
< N ( n ) <y

^
'n E

AF ( n ) bf ( n )

y < N ( n ) < 2y

^

+o

si

'+
y

N ( n ] s log N ( n )

(alog ( ))
(ylos (T))

^

si

(w)

(n)

^ Z llS AF bflogN | l
| ( n ) s + Cl

ncOp

(n )

'

Since si ( x ) satisfies si ( x ) C min (1,1) , we have

R <<

AF ( n ) | bf ( n )

| (

M

Relsl log N ( n ) log
y N (n )

)\

^

+0

( tl y
_

\

y

f N (n )

“

Re ( s ) +ci logN ( n )

J

(5.16)

Investigating the first term in (5.16), we see

z

AF ( n ) | bf ( n ) |

| ( )|

V|
< N ( n ) < 2y N ( n ) * logN ( n ) log

<y
«

y

1

i < Nl7) < 2y 1OSNW |1Og ( Nfc ) |
l

^^
TT

Z < |loS (
<
N ( p ) a 2y

1

JF ) |

N(

Letr = N ( tj ) a

y . Using the inequality log ( x + 1 ) >

estimate

) < ]y

^

(

log -N ( p ) a

L
<

N ( p ) a < 2y y

•

+1

for x > — 1 , we obtain the

Hence,

1
0

^

AF ( n ) |bf ( n ) |

M w)|

1

<—
ya

Z< <

2 r y
r O

1
y

Z 7 « y ffl°gy -

0< r<y

^

As for the second term in (5.16), by our choice of c -\ and Euler 's summation for-

64

’

mula, we get

AF bf
«y L
< y ‘' logy ,
.
Y
I
M ( ) + logN ( n )
y
N (n ) + °
« Fl n
(n)

—

(n )

~

1

a

a Ci

1 1 sy

ncOp

n

showing that
11 z
1 rci +iy
logL ( s + z, f ) — dz =
2m Jci i y
Z
"

Let c2 =

^

_ a+

•

AF ( n ) bf ( n )

N (n K y

N ( n ) s logN ( n ) +

Observe that if c2 L Re ( z )

0 (y fflogy ) .
-

(5.17)

c1 , we have Re ( s + z ) G Q, j ] .

By taking ao = \ in Lemma 5.2.1, we get
2n

log N ( 91)
logL ( s + z, f ) <

« log

\

Re ( s + z) -

(

N ( 9I )

(

^

^^

+ (|t| + y ) 2 + 3

uniformly for z with Re ( z ) y c2 and | Im ( z ) | y y .

logy

(5.18)

We now move the contour of integration in (5.17) to the line Re ( z ) = c2 . By our
assumption of the GRH, the integral only has a simple pole at z = 0. The residue
theorem then yields the following
1 fCi +iy
log L ( s + z, f ) — dz = log L ( s, f ) + £,
27ti J c i — iy
z

where
1 / rc 2-iy rc2 +iy rci +iy \
y!
dz.
£=—
logL ( s + z, f ) —
+
+
.
.
.
2m JCT -iy Jc 2-iy Jc2 + iy
J
\

I

65

Using (5.18), we observe that

J

logL ( s + z, f )

^

-

( \/u + ( |t| + y ) + 3 ) jlogyj yda

^ ^ ^ ynj
2

dz <C log
Cy

•

-1

^^ ^ ^

+ ci log

« y " log
“

c i +i y

2

(

N ( Ol )

N ( SK )

(

+ ( |t| + y ) 2 + 3

)

2n

+ ( |t| + yH + 3

.

j c +iy logL ( s + z, f ) — is treated similarly. For the second inte-

The third integral,

2

gral, we have

c;

rs
^

iog Lis + z' f )

dz

,ios Lis + cz + f i sT
^
’
< yC logy log ( NM ( i Tw + 3) )
f
T ( v/ +
+ + ynj logy
< VCz log (
ib )

*

db

'

^

2

y )2

3

v/k2 + (|t| + y ) 2 + 3)

^

N (9 )

log
C

<

^

k2

2

( | t|

ya— 2

Hence,
1

rci + iy

Mjcl _ llJl

^

)2

(

( logy ) 2

vz

sL ( s + 2' f ) Vd2 = l0sL ( s' f|

0

(

+0

yff - 2

as desired .

66

^

) ) ( lognj )

(

log N ( 9!) v/ k2 + ( |t| + y ) 2 + 3

Lemma 5.2.3. Let f e 17 ( 91) . Let y

^

^ 2 and s = a + it with a ^ 1 . Then

Proof . We have
10gLy ( S f ) /

Y_ AF bf = L

N (n ) < y

OO

(n ) (n )
N ( n ) s logn

ncOF

A ( n ) bf ( n )
N ( n ) * logN ( n )

P (« Ky

A(n )

(n )

bf
L N ( n ] logN
( )

=

s

ti

NCpKy
N ( p ) a >y

N

^

s

A ( n ) bf ( n )
N ( n ) * logN ( n )

1

L L N (p )

a 2

^

y°)
(
Z
V
<

>=

a3 2

<y
Since a

“

ff

a <T

N ( p ) >y a
— acr l 1

aa - 1

+2 .

1 , we get the desired result.

In the following proposition, we return to the task of estimating (5.10) .

Proposition 5.2.4. Let f
s = a + it with a

TLJOt ), and assume the GRH for L ( s, f ) . Let y > 2 and

1 . Then

(

( logy ) 2 ( log N ( 91)

log L ( s, f ) — log Ly ( s, f ) <

( v/k + ( |t| + y ) + 3) )
2

vy

67

2

2n

Proof . By Lemma 5.2.2, Lemma 5.2.3, and the triangle inequality, we have
llogLfs, f ) — log Ly ( s, f ) |

< logL ( s, f ) —
N (nKy

A ( n ) bf ( w )
10gLy ( S, f ) N ( n ) s logn +

( * 7 + t +y + ) )

( logq ) 2 ( log N ( K ) (

<

k2

- )2

(| |

3

N ( n ) <y

A ( n ) bf ( n )
N ( n ) slogN ( n )

2n

y° i

1

Vv

~

'

Since a > 1 , we obtain the desired bound .

In the following lemma, we provide an asymptotic upper bound for Ly ( s, f ) .
Lemma 5.2.5. Let 2

^ y x, and let s e C be such that Re ( s ) = 1 +
Ly ( s, f ) < ( logy ) 2 .

^

j

. Then, we have

Proof . Using (5.9 ), we write

lOg Ly ( S, f ) =

_Y ei

0f ( n )

N ( uKy

+ e —i 0 f ( P )

N IPY

“ eai6f ( p ] + e-ai6f ( p )

ZZ

+

aN ( p )

N ( P ) «Sy Q=2

pm

QS

(5.19)

£ N (p) a
~

+

L L aN ( p )

NfpKy a =1
p|Ot

.

as ’

Investigating the sums in (5.19) individually and using (5.7), we see that

Z
NfpKy

ei0f ( p ) _|_ e —i0f ( p )
N (p ) s

N(

pm

)

N ( JJ ) s '

pm

22 eQi0 f ( p ) _|_ e-ai0 f ( p )
aN ( tj )

a 2

z HP

NfpKy

pm

L
L
pKy =

=

00

2

L L= alSI ( ) aRe = 0(1 ),
^
pm
(s)

QS

N ( pKy a 2

68

and

£ N(p)

~

a

L L aN ( ) = 0 (1 )
^
QS

.

N ( p K y a =1
p \0 X

Now we can write

Ly ( s, f ) = exp

£

N (pKy

\ P\'^
which shows that

|V £

C, ( s, f ) « exp 2

N (pKy

^

1+ 0(1 ) ,

< exp ( 2 log logy ) = ( logy ) 2,

KV )

)

where we used [5][Equation Bl ] for an estimate of XN ( p ) < y jyqy

•

Now we are ready to prove the following important result from which Theorem
5.1.2 is deduced. In this result, we show that the partial sum

^

NfnKx

( n ) can be

approximated by the corresponding sum of Af ( n ) when restricted to ideals n with
P (n )

y for y < x.

Lemma 5.2.6. Let f e 11 ( 01) , and assume the GRHfor L ( s, f ) . Then for all real numbers

^

x and y such that ( logk9l ) 2 ( loglogk9I ) 8 < y < x

Y_ Af

N (n ) <x

(n) =

Y_ Af + o f
(n)

kN ( 9t ) we have

x ( logx ) ( logy ) 4 ( log ( kN ( 9t ) )

N (n ) <x
P( nKy

Vy

Proof . To simplify the exposition, we set

sfM = Y- Af ( n )

and

N (n ) <x

¥ ( x, y; Af ) =

Y_<x Af (") -

N (n)
P(n ) y

^

69

Without loss of generality, we assume that X G Z + ) . Let ci = 1 +

By Theorem

4.1.2 we have
Cl + i x

sw=

'

il

V

V

Ci

Xs
L ( s, f ) — ds + R,
s

—ix

where
R=

1
f

£<
<

N (n) x

+0

4C1 + XC1

y- |Af ( n ) | \

—

^

X

ncOF

I

N ( n ) Cl

Af ( n ) si (xlog
£
<
<

;/ /

V

V

x N ( n ) 2x

V

))

7 7

'

By the triangle inequality, we have

RV
71

| Af ( n ) |

£
5 < < * |l08 ( Nfe )

| Q

N ( n ) 2x

IA|( w ) |

L
|

|° ( )

< N ( n ) < 2x x l g

1+

o

4Cl + XC1
x

y |Af ( n )|

^— N ( n )

Cl

tlCOf

(y

lN ( n )

To estimate R, we use (5.4) to get

R<

y_

TF ( U )

| ( )|

|
< N ( n ) < 2x x los sfo

«

z

^F ( W )
1

+ lo§ x
ncOf N ( n )

f

+o

L

(y '
[ uhNW j

< N ( n ) < 2x X

(

| ( )

Observe that the first sum in (5.20) is equal to CF l +

70

(5.20)

1OS NW

) where CT is the Dedekind
/

zeta function of F. For any e > 0, the second sum in (5.20) is bounded as follows:

t

y

5 < N ( n ) <2x

TF ( n )
x lloff (

—^

I

1
I «X

N (n ) f

y
*

|
< N ( n ) < 2x

1

|IoS ( NTSI ) I
1

V—

L l (fife ) I

<< WS .

j < lN ( n ) < 2x 10§

-C 1_ 4y- Ir|
X

1 2

-

M <x

< x 2 logx C xe,
<

where |r| = N ( n ) — x, and the implied constant depends only on e . This yields

Sf ( x ) =

Cj

1
27

tilCl

+ix

-ix

Xs
L ( s, f ) — ds + Oe ( x ) .

(5.21)

s

Applying the argument we used to establish (5.21), we get
1

T ( x, y; Af ) = —

rci +ix

Jci — i x

xs
Ly ( s, f ) — ds + Oe ( xe ) .
s

To analyze the difference of Sf ( x ) and T ( x, y; Af ) , we set

Ry ( s, f ) =

b ( s, f )
Ly ( s, f ) '

We have

Sf ( x ) — T ( x, y; Af )
1

rC! + ix

i

rci + ix

^rj
J
—

S

( L ( s, f ) — Ly ( s, f ) ) — ds + 06 ( xe )
¥

s

Ly ( s, f ) ( exp ( log Ry ( s, f ) ) — 1 ) — ds

+ Oe ( xe ) .

(5.22)

71

By Proposition 5.2.4, we know that
N
( logy ) log ( N ( OT ) ( Vk + dt | + ) + 3 ) )
^
(s f ) 0
2

logRy

,

2

=

2n

ij 2

Vv

Observe that for kN ( 01) large enough, we have
( logy ) 2 log

(Vl ( Ol ) (Vk + ( |t| + y ) + 3)
2

2

vy
since kN ( OI )

2

8

^

= o(l )

^ x > y ^ ( logkN ( Ot ) ) (loglogkN ( OT ) ) . Hence, we can apply the

approximation ez = 1 + 0 ( | z| ) which is valid for \ z \ < 1 to get
1

( logy ) 2 log

exp ( logRy ( s , f ) ) = 1 + 0

(Vl ( Ot ) (Vk + ( |t| + y ) + 3)
2

2

Vy

^

(5.23)

for all s with Re ( s ) = ci and |Im ( s ) | + x. Applying Lemma 5.2.5 and (5.23) in (5.22)

gives

Sf ( x ) - ¥ ( x, y; Af ) <

( logy

) 4 xCl

Vy

x

L

(

Vk + ( |t| + y ) + 3) )2

log N ( 0I ) (

2

,

2

2n

-

dt

xlogx (logy ) 4 log ( kN ( Pt ) )
<
Vy
as desired .

The proof of Theorem 5.1.2 will follow as a result of Lemma 5.2.6 and the fol-

^^ .

lowing lemma, wherein we bound M X
Lemma 5.2.7. Let 10

TF )

y < x be real numbers . Then we have

¥ ( x, y ; TF ) < e ^ xlogx
72

log *
as u _ logy
—> oo .

Proof . We omit the proof of this lemma as it mirrors the proof of Lemma 4.3.7.
Finally, we are in position to prove Theorem 5.1.2.

Proof of Theorem 5.1 .2 . Supposex < kN ( 91) and choosey = ( logkN ( 91) ) 3. By Proposition 5.2.6

Y_ Af =

Y.

(n)

N ( nK*

N ( n ) <x

x ( logx ) ( log ( log kN ( 91) ) 3 ) 4 ( log ( kN ( 91) )

Af ( n ) + 0

P ( n ) s:( logkN 10r

( logkN ( hi ) ) 2

) )2

« ¥ ( x, ( logkN ( 91) ) 3; T2 ) + 0
< exp

- log *

^

x ( log x ) ( log ( log kN ( 91) ) 3 ) 4 ( log ( kN ( 91) )
( log kN ( 91) ) 2

xlogx + 0

2 log ( log kN ( 91) ) 3 )

= o ( xlogx ) .

( log ( 3 log kN ( 91) ) ) 4x log x
( log kN ( 91) ) 2

(5.24)

Now all that remains is to verify this asymptotic for x > kN ( 91) . To do this
we estimate the integral appearing in (5.21) . Via shifting the line of integration to

Re ( s ) =

\ + e we see

Ci + ix

1

ci — ix

/ r 2 + e + ix
Xs
L ( s, f ) — ds =
s

rci -ix

r 2 + e + ix \

yJ + e —ix J 2 + e —ix Jci +ix J

^

Xs
L ( s, f ) — ds = h — I2 — I3.
s

For j < Re ( s ) < 1, the sub-convexity bound given in Theorem 5.1.1 shows

log ( N ( 91) k2n ( | s + 3|2n ) ) 2 2 Ret \
loglog ( N ( 91) k2n ( |s | + 3 ) 2n )
~

L ( s, f ) < exp C

73

^1

'

Using this estimate, we get

^ ^
^

log ( N ( 9t ) k )

Ii < x 2 + e exp 2uC loglog ( N ( «n ) k )

( + e ) 2 + t2
|

log

lo

exp 2uC

logiog

v

)

(^ v ( i +

j .

e) )2

+3

log ( N (9t ) k ) \

< x 2 + e exp 2uC loglog ( N (9I ) k )
We also get
I2 «

’Ci

— ix
—-

2 +e ix

1
L ( - + e - ix, f
|L ( u

Xs

— ds =

/

1

,

x

:

j J ]/ ( l + ) +
2

£

dt

t2

'

'Cl

i +*

S

|L ( u — ix, f ) |

XU

/

2
\ u

+ X2

du

du = h + J 2 .
— ix, f ) | V uxu
2
+ x2

We have

C exp

<

^

/ 2nC log (Vu + x + 3 \
1
log ( N ( 9t ) k ) \ r 1
du
exp
2 x2
loglog ( N ( 0T] k ) 7 + e
2
2 3
Vu
+
x
log
log
Vu
( + +
y
2

Ji < xexp 2nC

2

Ji

)
jJ

log ( N ( 9t ) k )
loglog ( N ( <n ) k )

since v/u2 Tx2 x x. Moreover,
A
'
J 2 < xCl ' (log ( N ( 9t ) k2n ( |u - ix| + 3 ) 2n ))

j

xci

1
V u2

log ( N ( 94 ) k ( x + 5 ) ) ) C exp
xlogx (
2n

2n

<

^

+ x2
U

du

log ( N ( 9t ) k )
log log ( N ( 9rt ) k )

for some absolute positive constants A and C. The integral I3 can be estimated

similarly. Hence, we have

^^

n x

Af ( n ) < x 2 + eexp

log ( N ( 9rt ) k )
fa loglog
( N ( 9T ) k )
'

74

for some absolute positive constant Ci . In view of this asymptotic upper bound,
we see that under the GRH we have

(

p

^

Af ( u ) = o ( xlogx ) in the range x >

N ( nKx

j

logkN ( 9
exp c2 loglogkN
( Ol ) for some absolute positive constant C2 . In particular, the as-

sertion of Theorem 5.1.2 holds for x > kN ( 9t ) .

75

Chapter 6
Concluding Remarks
The work in this thesis can be extended in many ways. One could prove analogues

of Theorem 4.0.1 and Theorem 4.0.2 for the coefficients of symmetric m-th power
L-functions associated with modular forms. One immediate difficulty in this en-

deavour is finding a useful representation for Asymm ( n ) at prime powers.

Regarding the setting of Hilbert modular forms, one could pursue removing
the restriction on the class number of the underlying number field and consider

forms of non-parallel weight. This would require a good understanding of adelic
Hilbert modular forms. One could also explore proving analogues results for
the coefficients of symmetric power L-functions associated with Hilbert modular

forms.

In [16], there are additional results which we did not explore in this thesis.
These results are of a more probabilistic nature but could likewise be pursued in
the same direction as we have done here.

76

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79