Averages of fractional exponential sums weighted by Maass forms

Transcription

1 University of Iowa Iowa Research Online Theses and Dissertations Spring 2017 Averages of fractional exponential sums weighted by Maass forms Huan Qin University of Iowa Copyright 2017 Huan Qin This dissertation is available at Iowa Research Online: Recommended Citation Qin, Huan. "Averages of fractional exponential sums weighted by Maass forms." PhD Doctor of Philosophy thesis, University of Iowa, Follow this and additional works at: Part of the Mathematics Commons

2 AVERAGES OF FRACTIONA EXPONENTIA SUMS WEIGHTED BY MAASS FORMS by Huan Qin A thesis submitted in partial fulfillment of the requirements for the Doctor of Philosophy degree in Mathematics in the Graduate College of The University of Iowa May 2017 Thesis Supervisor: Professor Yangbo Ye

3 Graduate College The University of Iowa Iowa City, Iowa CERTIFICATE OF APPROVA PH.D. THESIS This is to certify that the Ph.D. thesis of Huan Qin has been approved by the Examining Committee for the thesis requirement for the Doctor of Philosophy degree in Mathematics at the May 2017 graduation. Thesis Committee: Prof. Yangbo Ye, Thesis Supervisor Prof. Philip Kutzko Prof. Muthu Krishnamurthy Prof. Gerhard Strohmer Prof. Mark McKee

4 ABSTRACT The purpose of this study is to investigate the oscillatory behavior of the fractional exponential sum weighted by certain automorphic forms for G 2 G 3 case. Automorphic forms are complex-values functions defined on some topological groups which satisfy a number of applicable properties. One nice property that all automorphic forms admit is the existence of Fourier series expansions, which allows us to study the properties of automorphic forms by investigating their corresponding Fourier coefficients. The Maass forms is one family of the classical automorphic forms, which is the major focus in this study. et f be a fixed Maass form for S 3 Z with Fourier coefficients A f m, n. Also, let {g j } be an orthonormal basis of the space of the Maass cusp form for S 2 Z with corresponding aplacian eigenvalues 1/4 + kj 2, k j > 0. For real α 0 and β > 0, we considered the asymptotics for the sum in the following form S X f g j, α, β = n=1 n A f m, nλ gj neαn β φ, 1 X where φ is a smooth function with compactly support, λ gj n denotes the n-th Fourier coefficient of g j, and X is a real parameter that tends to infinity. Also, ex = e 2πix throughout this thesis. We proved a bound of the weighted average sum of 1 over all aplacian eigenvalues, which is better than the trivial bound obtained by the classical Rankin- Selberg method. In this case, we allowed the form be vary so that we can obtain ii

5 information about their oscillatory behaviors in a different aspect. Similar to the proofs of the subconvexity bounds for Rankin-Selberg -functions for G 2 G 3 case, the method we used in this study includes several sophisticated techniques such as weighted first and second derivative test, Kutznetsov trace formula, and Voronoi summation formula. iii

6 PUBIC ABSTRACT During the last half-century, the theory of automorphic forms has become a major focus in the development of the modern number theory. Automorphic forms are functions from some topological groups to the complex plane, which have many applications to different aspects in Mathematics. Because automorphic forms have Fourier expansions. We can study its properties by studying the corresponding Fourier coefficients. Taking the weighted sums of these Fourier coefficients against various exponential functions will case a rise of resonance. We call this type of sum as a resonance sum. Resonance is a physical phenomenon that occurs between two interactive vibrating systems. Fixing one of these two vibrating systems, we may control the second one to detect the resonance frequencies of the first system, and thus obtain its oscillation spectrum. The most classical example of this is the Fourier series expansion of a periodic function, which is the resonance sum for G 1 case. This study is to learn the property of a resonance sum in a higher dimensional space. iv

7 TABE OF CONTENTS CHAPTER 1 INTRODUCTION History Development Basic Settings and the Main Results TECHNIQUES Derivative Tests Kutznetsov Trace Formula Voronoi s Summation Formula Stationary Phase Argument BOUNDS IN THE AVERAGE SUMS Weighted Summation Application of a Voronoi s Summation Formula Application of Derivative Tests REFERENCES v

8 1 CHAPTER 1 INTRODUCTION This thesis is to study the average of fractional exponential sums weighted by Maass forms for S 2 Z and S 3 Z. In the first section of this chapter, the historical development in the subconvexity bound problem of automorphic -function will be brushed up on, which also serves as an motivation of this research. Moreover, basic settings of the whole problem including the definitions and notations that will be used throughout the whole thesis, as well as the main result will be introduced in the next section. This work is largely inspired by Sarnark [30], iu-ye [18], i [17], and McKee- Sun-Ye [20], which are all important works for finding asymptotic bounds for certain type of Rankin-Selberg -functions. It is worth of mentioning several sophisticated techniques that I have applied to this study that were adapted from those papers. In each of the fours sections in Chapter 2, the derivative tests, the Kutznetsov trace formula, the Voronoi s summation formula, and the stationary phase argument will be introduced. In Chapter 3, we provide a proof of out main result.

9 2 1.1 History Development The first and most famous -function has been given Riemann s name. The Riemann-Zeta function is defined by the generalized harmonic series ζs = n=1 1 n s, where s = σ + it and σ > 1. Even though we name this function after Riemann s name, it was studied even before Euler. The start of the branch of analytic number theory has been considered as Euler s discovery of connections between ζs and the prime numbers, which is described in Euler product formula: ζs = p 1 1 p s 1. Euler published his results in his widely know Introductio, and it was this work that Riemann went on from. Riemann stated that ζs has a meromorphic continuation to the whole complex plane. There is only one singularity, which is a simple pole with residue 1 at s = 1. Moreover, this analytic continuation satisfied the following functional equation ζ1 s = π s 2 s 2 Γs cosπs ζs, 2 for all complex numbers, provided valid gamma functions. These results are in Riemann s only number theory publication Über die Anzahl der Primzahlen unter einer

10 3 gegebenen Größe On the number of primes that are less than a given quantity in It was his way of thanking the Berlin Academy for appointing him a corresponding member. This paper was great, not only because Riemann brought the famous Riemann hypothesis in it. Before this paper, no one would, at first glance, expect to connect the natural numbers, which represents the discrete and the disconnected aspect of mathematics to complex analysis, which has to do with the continuous representations and treatments. We know that ζs has trivial zeros at 2n for n 1, which are all negative even integers. However, the trivial zeros are not the only values for which ζs is zero. The others zeros are called non-trivial zeros. Here comes the widely known Riemann Hypothesis: All non-trivial zeros of ζs are on the critical line: Res = 1 2. The hypothesis in Riemann s words: and it is likely that all roots have real part 1/2: Of course, it would be desirable to have a rigorous proof of this; in the meantime, after a few perfunctory vain attempts, I temporarily put aside looking for one, for it seemed unnecessary for the next objective of my investigation. Even though Riemann did not continue to work on his own hypothesis, many accomplished mathematicians such as Stieltjes, Hardy, and Ramanujan, have tried to prove it. indelöf hypothesis is one consequence of the Riemann hypothesis investigaing

11 4 the rate of growth of ζs on the critical line, which states that for any ε > 0, ζ it t + 1 ε as t. This result was proved by Finnish mathematician Ernst eonard indelöf in This hypothesis says that: for any ε > 0, as t tends to infinity, the number of zeros of ζs with real part be at least 1 + ε and imaginary part be between t and 2 t + 1 is olog t. The Riemann hypothesis implies that there are no zeros at all in this region and thus implies the indelöf hypothesis. One may think that the indelöf hypothesis should be easier to establish, but most number theorists think that the Riemann hypothesis must be solved first. The point is that the Riemann hypothesis is constructed in a more natural mathematical way. In the general theory of -functions, there are some quite significant features such as conductor and of primitivity do not show in the fundamental Riemann- Zeta Function. So, we need to consider one kind of more generalized -function, which is called the Dirichlet -function and has been defined as the Dirichlet series associated to Dirichlet characters χ as s, χ = n=1 χn n s = p 1 χp 1. p s Just like the case in ζs, the above Dirichlet series and the corresponding Euler product are absolutely convergent for Res > 1; the Dirichlet -function has analytic continuation to the whole complex plane and satisfies a functional equation; it is also

12 5 conjectured to obey the generalized Riemann hypothesis. Here, we can follow Iwaniec-Kowalski [8] to provide a definition for -functions: we say that s, f is an -function if we have the following data and conditions. 1 A Dirichlet series with Euler product of degree d 1, s, f = n=1 λ f n n s = p 1 α 1 1p 1 α 1 dp p s p s with λ f 1 = 1, λ f n C. Of course, the series and Euler products must be absolutely convergent for Res > 1. The α i p, 1 i d, are called the local roots or local parameters of s, f at p with the property that for all p, α i p < p. 2 A gamma factor γs, f = π ds/2 d s + κj Γ 2 j=1 where the numbers κ j C are called the local parameters of s, f at infinity. We assume these numbers are either real or show up in conjugate pairs. In addition, Rek j > 1, which means that γs, f has no zero in C and no pole for Res 1. 3 An integer qf 1, called the conductor of s, f, such that α i p 0 1 i d for p qf and such prime p is said to be unramified. As a generalization of Riemann-zeta function and the Dirichlet -function,

13 6 anglands considered a large family of -functions defined as the following form s, π = n=1 λ π n n s for Res > 1, where π is the automorphic cuspidal representations of G n A Q the general linear group G n over the adele ring of Q, λ π n is the Fourier coefficient of π, with nice family structures. The nice relationship among this family of - functions is called anglands Functoriality Conjecture, which means the principle that sometimes we can use existing -functions to build a new on by taking operations on their Fourier coefficients λ π n, such as raising to the squared power as λ π n 2 or λ π n 2. However, it is not known whether or not the resulting representation is again an automorphic one, although this is suspected to be true and is known in a few select cases. That is why this remains as a conjecture. The main conjectures in the anglands Program predict the existence of a correspondence between analytic number theory and algebraic number theory. Specifically, s, π s, ρ, which means every -function arising from an automorphic cuspidal representation of G n A Q is equal to an Artin -function arising from a finite dimensional representations of Galois group of a number field. The analytic properties of the left hand side has been well studied. So, the anglands program has been considered as an effective tool for finding a non-abelian class field theory.

14 7 The Grand Riemann Hypothesis refers to the following conjectural statement about zeros of the family of -functions defined above: All non-trivial zeros of s, π are on the critical line: Res = 1 2. There is no need to say we believe that every such -function satisfies the Grand Riemann Hypothesis. However, proving this even for one -function would be an far-reaching achievement in the history of human beings. For this generalized case, we also want to investigate the size of an -function at its central point. Thus, we expect the generalized indelöf hypothesis to follow from the generalizedgrand Riemann hypothesis as in the Riemann-zeta function case. However, this remains as an open question. For any ε > 0, it, π 1 + t + Q π ε where Q π is a conductor of the representation π. By Applying the Phragmén-indelöf Principle in complex analysis, we can obtain a trivial bound of it, π as follows: it, π 1 + t + Q π 1 4 +ε. This trivial upper bound is also called the convexity bound. The achievable goal is to prove an upper bound for it, π that better than the convexity bound. And this problem is called the subconvexity problem of automorphic -functions

15 8 Here are some known results. The recent breakthrough was obtained by Xiaoqing i [17]. In her paper published in Annals of Mathematics, she proved a subconvexity bound for the -function in G 2 Z G 3 Z case.the result has been improved by my advisor YangboYe with two colleges Mark McKee and Haiwei Sun [20] in their paper published in Transactions of the AMS. This present research is to study the oscillatory behavior of Fourier coefficients λ π n. My goal is to find an upper bound in average for a sum related to the subconvexity bound of automorphic -functions. Here are the basic settings and the main results. 1.2 Basic Settings and the Main Results et f be a fixed Maass form for S 3 Z with Fourier coefficients A f m, n. et g be a Maass cusp form for S 2 Z with aplacian eigenvalue 1/4 + k 2, k > 0. et {g j } be an othogonormal basis of the space of Maass cup forms with aplacian eigenvalue 1/4 + kj 2, k j > 0. Denote the n-th Fourier coefficient of g j by λ gj n, normalized by < g j, g j >= 1. Then, g j has the Fourier-Whittaker expansion g j z = y cosh πk j 1 2 λ gj nk ikj 2π n yenx. n 0 Here, K ikj is the Bessel K-function, z = x + iy, and ex = e 2πix. Since each g j is normalized by g j = 1, the leading coefficient λ gj 1 is no longer equal to 1. By Hoffsein-ockhat, we can use a new normalization of g j with λ gj 1 = 1 after a Ok ɛ j

16 9 discrepancy. We consider the following resonance sum: S X f g, α, β = n A f 1, nλ g neαn β φ X n>0 1.1 where α 0, β > 0, φ C c 1, 2, X is a parameter tends to infinity. Motivated by wokrs in iu-ye [18], Ren-Ye [27], i [17] and Salazar-Ye [29], we will provide a non-trivial bound for the weighted summation of S X f g j over K k j K + when β = 1/3. Specifically, we have the following theorem: Theorem 1.1. Suppose α 0 is fixed. et {g j } be an orthonormal basis of Hecke- Maass eigenforms with aplace eigenvalues 1/4 + k 2 j, k j > 0. et K be a parameter tending towards infinity with K ε K 1 ε and K X. Then, for β = 1/3, K k j K+ e k j 2 K 2 S X f g j, α, 1 3 K 1+ε X 1 2 +ε + K ε X 1+ε. 1.2 Note that under the assumption K X, the above summation 1.2 is dominated by K 1+ε X 1 2 +ε. Otherwise, it is dominated by K ε X 1+ε. Since by Weyl s law, there are about 1+ε K terms in the summation, Rankin-Selberg method yields the trivial bound 1+ε KX 1+ɛ. This problem was first brought by Iwaniec-uo-Sarnak [9],they studied a resonance sum with f being a holomorphic cusp form in S 2 Z, β = 1, and α = ±2 q q Z +. Followed their works, Ren-Ye [23] and Sun [32] investigated the resonance behavior of automorphic forms for certain α and β in G 2 Z. They were able to

17 10 obtain the main term and thus asymptotic upper bound of the corresponding sum. ater, in Sun-Wu [33], the authors considered the case when f is a Maass cusp form for S 2 Z, similar asymptotic results can be obtained. As a crucial tool for future works in G 3 Z cases, Ren-Ye [24] proved the asymptotic Voronoi s summation formula for S 3 Z and discussed its applications and the duality property.

18 11 CHAPTER 2 TECHNIQUES It is worth mentioning several sophisticated techniques that I have applied to this present research. For dealing with the exponential integral by either showing it is negligible or obtaining the asymptotic expansions, the second derivative test in Huxley[6] is used to bound the integral; a weighted first derivative test from MacKee- Sun-Ye [20] with more strength than the usual first derivative test in Huxley [6] is used to deal with the instances when the phase function is infinitely differentiable. The Kuznetsov trace formula is one of the various methods that can be used to develop the spectral decompositions of sums of Kloosterman sums. After applying the Kuznetsov trace formula and manipulating the order of terms, the desired sum can be rewritten as a separate spectral part and a geometric part. In particular, the integral on the spectral side represents the continuous spectrum of the aplace operator with the divisor function being the Fourier coefficient of Eisenstein Series. Another very important and widely used technique in modern analytic number theory is the Voronoi s summation formula. It is used to convert certain sums of arithmetic terms into sums over integrals. An asymptotic expansion of Voronoi s summations formula for Maass forms in S 3 Z proved by Ren-Ye [24] is used in this research. Finally, the stationary phase argument technique is used to control the growth of these obtained integrals.

19 Derivative Tests Consider one kind of integrals named exponential integrals of the following form β α gxefx dx, 2.1 where x, fx, and gx are all real, f x changes signs at some point between α and β. Here, the function fx is called the exponent and the gx is called the weight. We will need to use the second derivative test lemma in Huxley[6] to obtain its asymptotic expansion. The lemma is state here in its entirety. emma 2.1. Huxley, [6] et fx be real and twice differentiable on the open interval α, β with f x λ > 0 on α, β. et gx be real, and V be the total variation of gx on the closed interval [α, β] plus the maximum modulus of gx on [α, β]. Then β α gxefx dx 4V πλ. While in some other cases, we need to show the exponential integrals 2.1 are negligible. The weighted first derivative test lemma in Huxley [6] was widely used to accomplish this task. emma 2.2. Huxley, [6] et fx be a real function, three times continuously differentiable for α x β, and let gx be a real function, twice continously differentiable

20 13 for α x β. Suppose that there are positive parameters M, N, T, U with M β α, and positive constants C, such that, for α x β, f r x Cr T/M r, g s x Cs U/N s, for r = 2, 3, and s = 0, 1, 2. If f x and f x do not change sign on the interval [α, β], then we have I = β α gxefx dx = gβefβ gαefα 2πif β 2πif α T U + O 1 + MN M + M 2 min f x 1 2 N 2 T/M min f x 3 The implies constants are constructed from the constants C r. In McKee-Sun-Ye [20], the authors proved a stronger version of the above emma 2.2 by including more boundary terms and smaller error terms. Here is the weighted first derivative test theorem in McKee-Sun-Ye[20]: emma 2.3. McKee, Sun and Ye, [20] et σt be a real-valued function, n + 2 times continuously differentiable for a t b, and let gt be a real-valued function n + 1 times continuously differentiable for a t b. Suppose that there are positive parameters M, N, T, U with M b a, and positive constants C r such that for a t b, σ r t C r T M r, gs t C s U N s, for r = 2,..., n + 2 and s = 0,..., n + 1. If σ t and σ t do not change signs on

21 14 the interval [a, b], then we have b a eσtgtdt = [ eσt j=1 n H i t i=1 ] b a +O M [n/2] UT j N min σ n+j+1 M 2j j=1 M +O N + 1 U N n min σ n+1 n UT j n j +O min σ n+j+1 M 2j t=0 n j t=j 1 N n j t M t 1, N n j t M t where H 1 t = gt 2πiσ t, H it = H i 1t 2πiσ t for i = 2,..., n. Besides, we need to use the following lemma theorem 1.2 from Salazar-Ye [29] to deal with an oscillatory integral of the form W K, x = R tk e + x πt 2π cosh huu + K tdt, where hr is assumed to be an even analytic function on the strip Imr 1/2 + ε and hr r 2 δ for some δ > 0 as r. Also, assume hr 0. Thus, hr is a Schwartz function on R and is negligible outside ε, ε. emma 2.4. Salazar-Ye, [29] Suppose K ε K 1 ε. i If x < K 1 ε/2, then W K, M K M for any M > 0.

22 15 ii If x K 1 ε/2, then W K, x = n K 2n+5 K 2n+2 W ν x + O + O. 2.2 ν=0 x n+2 x n+1 Here W ν x = iν 2ν 1!!1 + i K 2ν+1 sgnx ν π 2ν+1/2 4K 2 + x 2 H 2νγ 2.3 ν/2+1/4 sgnx e 4K2 + x 2π 2 K 2K π sinh 1, x where H 2ν γ is defined as H 2ν γ = H2νγ 2ν! + 2ν 1 l=0 3 n 1,...,n k 2n+3 n 1 + +n k =2ν l+2k H lγ l! 2ν l k=1 2 k C 2ν,l,k σ 2 γ σ n1 γ σ nk γ, 2.4 n 1! n k! with γ = η 1 π sinh 1 Kc 2π t 3 X. 2.5 Hz = hu u K + 1 z, 2.6 σz = zk + 2η 1 t3 X πz cosh c 2.7

23 16 and C 2ν,l,k are constants. Thoughout this thesis, ε is any arbitrarily small positive number. Its value may be different on each occurrence. 2.2 Kutznetsov Trace Formula Use the same notation as in the basic settings in section 1.2, and let {g j } be an othonormal basis of the space of Maass cup forms with aplacian eigenvalue 1/4 + kj 2, k j > 0. Denote the n-th Fourier coefficient of g j by λ gj n, normalized by < g j, g j >= 1. The Fourier expansion of g j is of the form g j z = y cosh πk j 1 2 λ gj nk ikj 2π n yenx. n 0 This normalization for g j is very important. In fact, the Kuznetsov formula will not be valid for Maass form case if {g j } is not a orthonormal base. The Kuznetsov trace formula is stated as follows: emma 2.5. Kuznetsov, [13] et hr satisfies the following conditions hr is even. hr is holomorphic in the strip Imr ε. hr r δ in the strip.

24 17 Then, for any n, m 1, hk j λ gj mλ gj n τ ir nτ ir mhr 2 dr 2.8 π g R j ζ1 + 2ir = δ n,m tanhπrhrdr 2.9 π 2 R + 2i Sn, m; c 4π mn hrr J 2ir dr, 2.10 π c c coshπr c 1 R where τ ν n = ab= n a/bν and Sn, m; c is the classical Kloomsterman sum. The left-hand side 2.8 of the formula contains the spectral information, and the right-hand side 2.9, 2.10 represent the geometric aspect. Moreover, the integral in term 2.8 represents the continuous spectrum of the aplacian operate. And the divisor function τ ir n is the Fourier coefficient of corresponding Eisenstein series. The following ζ1 + 2ir 1 log2 + r is called a bound of de la Vallée Poussin. Because of this bound, the integral on the left-hand side 2.8 converges absolutely. Since τ ir n n and hr r δ, the integral on the right-hand side in term 2.9 is also convergent. The Kloomsterman sums Sn, m; c is one very important type of arithmetic function in analytic number theory because it links with spectral theory of automorphic forms. The classical Kloomsterman sum is defined by Sn, m; c = x mod c nx + m x e, c

25 18 for integers n, m, and c 1. The here means the summation is taken over x such that x x 1mod c. The Kloomsterman sums satisfy nice symmetric properties: Sn, m; c = Sm, n; c and if n, c = 1, Snn, m; c = Sn, mn ; c. Also, it satisfy a nice multiplicative property: if c, d = 1, Sn, m; cd = Sn c, m c; dsn d, m d; c. Kloomsterman sums will make appearances in the proof of the main theorem. They has close connection with the spectral theory of automorphic forms. 2.3 Voronoi s Summation Formula This section is to present an important tool, the Voronoi s summation formula for G 3 case. Follow Goldfield -i [5], McKee-Sun-Ye [20], and Ren-Ye [27], consider a Maass form f for S 3 Z of type ν = ν 1, ν 2. Then, we can define the angland s parameters for f as µ f 1 = ν 1 + 2ν 2 1, µ f 2 = ν 1 ν 2, µ f 3 = 1 2ν 1 ν 2.

26 19 Moreover, f has the following Fourier-Whittaker expansion: fz = γ U 2 Z\S 2 Z m 1 1 m 2 0 A f m 1, m 2 W J m 1 m 2 M γ 0 z, ν, ψ 1,1, 0 1 γ where U 2 =, W J is the Jacquet-Whittaker function, M = diagm 1 m 2, m 1, 1, 0 1 ψ 1,1 is a fixed character on the abelianization of the standard unipotent upper triangular subgroup of S 3 Z, and A f m 1, m 2 are Fourier coefficients of f. et ψx C c 0, and denote ψs be the Mellin transform as ψs = 0 ψxx s 1 dx. For k = 0 or 1, define Ψ k x = Res=σ 3 Γ π 3 x s j=1 1+s+2k+µf j Γ 2 s+µ f j 2 ψ s kds, where σ > max 1 Reµ f 1 2k, 1 Reµ f 2 2k, 1 Reµ f 3 2k. and denote Ψ 0 0,1x = Ψ 0 x + 1 iπ 3 x Ψ 1x, Ψ 1 0,1x = Ψ 0 x 1 iπ 3 x Ψ 1x.

27 20 The Voronoi s summation formula is stated in the following lemma. emma 2.6. Miller-Schmit, [22] et A f m, n be the Fourier coefficients of the Maass cusp form f for S 3 Z. Suppose that ψ C c 0,. et c, d be integers such that c 1, c, d = 1 and d d 1mod c. Then n d A f m, ne ψn c n>0 = cπ 5/2 4i + cπ 5/2 4i n 1 cm n 2 >0 n 1 cm n 2 >0 A f n 2, n 1 S n 1 n 2 A f n 1, n 2 S n 1 n 2 md, n 2 ; mc n 1 md, n 2 ; mc n 1 Ψ 0 0,1 Ψ 1 0,1 n2 n 2 1 c 3 m n2 n 2 1 c 3 m, where again Sa, b; r is the classical Kloomsterman sum. Since Ψ 1 x/x has very similar asymptotic behavior as Ψ 0, we only need to deal the case with Ψ 0. The asymptotics of Ψ 0 x are included in the following lemma. emma 2.7. Ren-Ye, [27] Suppose that ψ is a fixed smooth function of compact support on [X, 2X] where X > 0. Then for x > 0, xx 1, K 1, we have Ψ 0 x = 2π 3 xi +O 0 ψy xx 2 K 3 K j=1, c j cos6πxy 1/3 + d j sin6πxy 1/3 dy xy j/3 where c j and d j are constants with c 1 = 0 and d 1 = 2/ 3π.

28 Stationary Phase Argument Similar to the weighted first derivative test we mentioned in section 2.1, in McKee-Sun-Ye [20], the authors improved the traditional weighted stationary phase integral lemma in Huxley [6]. We stated both statements in their entirety below for comparison. emma 2.8. Huxley, [6] et fx be a real function, four times continuously differentiable for α x β, and let gx be a real function, three times continuously differentiable for α x β. Suppose that there are positive parameters M, N, T, U, with M β α, N M/ T, and positive constants C r such that, for α x β, f r x Cr T/M r, g s x Cs U/N s, for r = 2, 3, 4 and s = 0, 1, 2, 3, and f x T/C 2 M 2. Suppose also that f x changes sign from negative to positive at a point x = γ with

29 22 α < γ < β. If T is sufficiently large in terms of the constants C r, then we have β = α +O +O gxefxdx gγefγ + 1/8 f γ M 4 U T 2 MU T 3/2 1 + M N 1 + M 2. N + gβefβ gαefα 2πif β 2πif α 2 1 γ α β γ 3 The implies constants are constructed from the constants C r. emma 2.9. McKee-Sun-Ye, [20] et σt be a real-valued function 2n + 3 times continuously differentiable for a t b, and let gt be a real-valued function 2n + 1 times continuously differentiable for a t b. Suppose σ t changes signs only at t = γ, from negative to positive, with a < γ < b. Define H k t by H 1 t = gt 2πiσ t, H it = H i 1t 2πiσ t for i = 2,..., n. Suppose that there are positive parameters M, N, T, U with M > b a and positive constants C r such that for a t b, σ r t T Cr M, r σ t T C 2 M, 2 g s t Cs U N s

30 23 for r = 2,..., 2n + 3 and s = 0,..., 2n + 1. Define log 2 = min, C 2 C max {C k}. 2 k 2n+3 If T is sufficiently large such that T 1 2n+3 > 1, we have for n 2 that = b a gteσtdt [ eσγ + 1/8 n 1 j 2j 1!! n+1 gγ + ϖ 2j + eσt H σ γ 4πiλ j=1 2 j i t i=1 UM 2n+5 1 +O T n+2 N n+2 γ a + 1 n+2 b γ n+2 UM 2n+4 1 +O T n+2 γ a + 1 2n+3 b γ 2n+3 UM 2n+4 1 +O T n+2 N 2n γ a b γ 3 U M 2n+2 +O T n+1 N + M. 2n+1 ] b a Here ϖ 2ν = η 2ν + η l = gl γ l! λ l = σl γ l! 2ν 1 l=0 2ν l η l k=1 C 2ν,l,k λ k 2 for 0 l 2n, for 2 l 2n + 2, 3 n 1,...,n k 2n+3 n 1 + +n k =2ν l+2k λ n1 λ nk, and C 2ν,l,k are absolute constants.

31 24 CHAPTER 3 BOUNDS IN THE AVERAGE SUMS In this chapter, we will prove the main result Theorem Weighted Summation Suppose K ε = K u K 1 ε and K. et hr satisfies the conditions presented in emma 2.5, i.e. hr is an even holomorphic function in the strip Imr 1/2 + ε and hr r + 1 N for arbitrary N > 0. We will estimate the bound for the weighted summation of S X f g j, α, β in 1.1, which is of the form: g j [ kj K h + h k ] j + K S X f g j, α, β. 3.1 First, we rewrite the S X f g j, α, β with the right-hand side of its definition in 1.1. Thus, the weighted summation goes to n [ A f n, 1eαn β kj K φ h X n g j + h k ] j + K λ gj n. 3.2 With the defined normalization λ gj 1 = 1, we then apply the Kuznetsov trace

32 25 formula to the inner sum of the above expression 3.2: g j = δ 1,n π 2 [ kj K h 2iπ 1 c 1 + h k ] j + K λ gj n λ gj 1 [ r K h + h r + K ] tanhπr rdr 3.3 R 1 [ r K d ir nd ir 1 h + h r + K ] 1 dr 3.4 π R ζ1 + 2ir 2 [ 4π n r K + J 2ir h + h r + K ] r R c coshπr dr Sn, 1; c, 3.5 c where d ν n = ab= n a/bν is the division function, ζs is the Riemann zeta function, and Sn, m; c is the classical Kloosterman sum. For term 3.4, we break the integral into two: 1 π 1 π R R r K 1 d ir nd ir 1h dr 3.6 ζ1 + 2ir 2 d ir nd ir 1h r + K 1 dr 3.7 ζ1 + 2ir 2 For the first integral 3.6, let u = r K By d ir = d ir and ζ s = ζs, the term 3.4 goes to ; and for the second one 3.7, let u = r+k. 2 π R hu d iu+k nd iu+k 1 ζ1 + 2iu + K du. 2 Note that ζ1 + 2ir c log 2/3 2 + r for some c > 0. Term 3.4 contributes to

33 26 the weighted summation 3.1 as 2 π R n n A f n, 1φ d iu+k neαn β 3.8 X hu ζ1 + 2iu + K du, 2 which is asymptotically bounded by K ε X 1+ε by the Rankin-Selberg methods. Now, we turn to the diagonal term 3.3. Since it is nonzero only if n = 1, this diagonal term contributes to the sum 3.1 as 1 1 D := A f 1, 1eαφ X π [ 2 r K tanhπr h R + h r + K ] rdr. 3.9 However, D = 0 since φ 1 X = 0, as X. For finding the asymptotics for the off-diagonal term 3.5, we follow iu-ye

34 27 [18] closely. et x = 4π n c and define [ r K V K, x := J 2ir x h + h r + K ] r R coshπr dr = 1 [ r K J 2ir x h + h r + K ] r 2 R coshπr dr 1 [ r K J 2ir x h + h r + K ] r 2 R coshπr dr = 1 [J 2ir x J 2ir x] 2 R [ r K h + h r + K ] r coshπr dr = 1 J 2ir x J 2ir x 2 R sinhx [ r K h + h r + K ] tanhπrrdr The assumptions on hr are equivalent to state that the hr is negligible for r outside K 1+ε, K + 1+ε and also for r outside K 1+ε, K + 1+ε. Since the h functions on the right-hand side of 3.10 isolate r to ±K, we can remove tanhπr by getting a negligible OK N for any N > 0. By Parseval s theorem, 3.10 goes to J2ir x J 2ir x y 3.11 V K, x = 1 2 R [ h sinhx r K +OK N. r + h r + K r ] ydy

35 28 Then, we apply the following formula in Bateman[1] to 3.11 J2ir x J 2ir x y = i cosx coshπy. sinhx With a variable change from y to y in 3.11, we have V K, x = 1 2i cosx coshπy 3.12 R [ r K h r + h r + K r ] ydy +OK N. Then, we write out the Fourier transforms of [h + h] term in 3.12 explicitly. = = [ r K h r + h r + K r ] y r K h r erydr + h r + K r erydr R R hu u + K e u + Ky du R + hu u + k e u + Ky du R = e Kyhu u + K y +ekyhu u + K y, where we did variable changes by letting u = ± r K. Since K1 ε for some ε > 0

36 29 and hu isolates u to O1, we may drop the absolute value. Thus, 3.12 goes to V K, x = 1 cosx coshπy 2i R [e Ky huu + K y +eky huu + K y]dy +OK N. etting y = t/, V K, x = 1 πt cos x cosh 2i R [ e Kt huu + K t Kt ] +e huu + K t dt πt = i cos x cosh R tk e huu + K tdt 3.13 = 1 2i W K,x + W K, x, where W K, x = := R R tk e + x πt 2π cosh huu + K tdt 3.14 eσtgtdt.

37 Application of a Voronoi s Summation Formula et η 1, η 2 and η 3 be independently either 1 or 1. By emma 2.4 in section 2.1, we only need to consider the case when x K 1 ε/2 in Recall that we have assumed x = 4π n c when we define V K, 3.10 in the previous section. Combine these two restrictions for x, we get c 4π n/k 1 ε/2. Moreover, since φ C c 1, 2 in 3.2, 1 n X 2 = c 4 2Xπ K 1 ε/2. By opening up the Kloosterman sum, a typical term in 3.5 is of the following form R η 1 ν = K 2ν+1 ν c c 4 2X z mod c K 1 ε/2 z,c=1 n A f 1, neαn β φ X n nz e c 2 K 2 + 4π 2 n ν c H 2ν η 1 Kc π sinh 1 2π n η1 e c2 K πc 2 + 4π 2 n η 1K π z e c sinh 1 Kc 2π. n 3.15 Next, we apply the Voronoi s summation formula for G 3 case we presented in emma 2.6 to the summation over n in 3.15 with y ψy = eαy β φ c 2 K 2 + 4π 2 y ν X H 2ν η 1 Kc sinh 1 e π 2π y η1 c2 K πc 2 + 4π 2 y η 1K π sinh 1 Kc 2π. y

38 31 This summation goes to A f 1, ne n = cπ 5/2 4i + cπ 5/2 4i n 1 c n 2 >0 n 1 c n 2 >0 nz ψn 3.17 c c n 2 1 n 2 A f n 2, n 1 S n 1 n 2 A f n 1, n 2 S n 1 n 2 z, n 2 ; n 1 z, n 2 ; Ψ 0 0,1 c n 1 Ψ 1 0,1 c 3 n 2 1 n 2 c 3, where for k = 0, 1, Ψ k 0,1x = Ψ 0 x + 1 k 1 xπ 3 i Ψ 1x. And by emma 2.7 in section 2.3, for any fixed r 1, x = n 2 n 2 1/c 3, and xx 1, Ψ 0 x = 2π 3 xi +O = π 3 r j=1 +O = π 3 r j=1 0 ψy xx 2 r 3 x 1 j 3 0 xx 2 r 3 x 1 j 3 0 H 2ν η 1 π e η1 r j=1 c j cos6πxy 1/3 + d j sin6πxy 1/3 dy xy j/3 [ ] ψy ic j + d j e3xy icj d j e 3xy 1 dy 3 y j/3 y c eαy β φ 2 K 2 + 4π 2 y ν X Kc sinh 1 2π y c2 K 2 + 4π 2 y η 1K π sinh 1 Kc 2π y πc [ ] ic j + d j e3xy icj d j e 3xy 1 dy 3 +O xx 2 r 3 y j/3 3.18

39 32 Recall that c j and d j are constants depending on the anglands parameters with c 1 = 0 and d 1 = 2/ 3π. Note that by doing change of variables, x 1 Ψ 1 x has similar asymptotic to Ψ 0, so we only need to deal with Ψ 0. et a j = ic j + d j, b j = ic j d j, and set y = t 3 X with dy = 3t 2 Xdt. For x = n 2 n 2 1/c 3, 3.18 goes to r Ψ 0 x = 3π 3 x 1 j 3 X ν 2 j j=1 2 1/3 2 ν 2 ck 1 4 φt 3 X + 4π 2 t 3 1 H 2ν η 1 Kc π sinh 1 2π t 3 X e η 2 1 X ck X + 4π πc 2 t 3 η 1K Kc π sinh 1 2π t 3 X ] eαx β t 3β [a j e3xt 3 X bj e 3xt 3 X 1 3 t 2 j dt +O xx 2 r 3 r = 3π 3 x 1 j 3 X ν 2 j j=1 +O xx 2 r 3 [ aj I η 1η 3 + j + b j I η 1η 3 ] j. 3.19

40 33 Here, we define I η 1η 2 η 3 j t := 2 1/3 1 2 ck t 2 j φt 3 X + 4π 2 t 3 ν H 2ν η 1 Kc π sinh 1 2π t 3 X e η 2 1 X ck X + 4π πc 2 t 3 η 1K Kc π sinh 1 2π t 3 X 3.20 e η 3 α X β t 3β + 3η 2 xx 1/3 t dt. = 2 1/3 1 G j teθtdt, where 2 ck G j t := t 2 j φt 3 X + 4π 2 t 3 ν H 2ν γ, 3.21 and θt = η 3 α X β t 3β + 3η 2 xx 1/3 t + η 2 1 X ck X + 4π πc 2 t 3 η 1K Kc π sinh 1 2π t 3 X Recall that H 2ν γ is defined as 2.4 in emma 2.4 and x = n 2 n 2 1/c 3 is from Application of Derivative Tests In order to apply derivative tests to the integrals in 3.20, we need to bound derivatives of G j t and θt in order to satisfy assumptions of derivative tests. We

41 34 will deal with G j t and θt separately under the assumption in the main result in Theorem 1.1, which is K X. First, we consider G j t. Since 1 t 2 1/3, t 2 j s 1 and φt 3 s 1. Moreover, since c 4 2Xπ K 1 ε/2, ck X 4 2πK ε 2 4 2π. Thus, ck X 2 + 4π 2 t 3 1 X, which implies 2 ck X + 4π 2 t 3 ν s X ν , for s 0. Now, we compute the derivatives of H 2ν γ s. The first term in H 2ν γ defined in 2.4, H 2ν γ/2ν! is negligible since Ht function defined in 2.6 is of rapid decay, i.e. derivative: d s dz s H l z 1. Also the term γ defined in 2.5 has the first γ = 3η 1 Kc X 1 2πt ck X 2 + 4π2 t 3 Kc X, and the higher derivatives of γ are of the same size. Thus, by the chain rule, d s s Kc dt s Hl γ. X

42 35 Then, in H 2ν γ defined in 2.4, there is a sum of products in H 2ν γ of the form 2ν 1 l=0 H lγ l! 2ν l k=1 2 k C 2ν,l,k σ 2 γ 3 n 1,...,n k 2n+3 n 1 + +n k =2ν l+2k σ n1 γ σ nk γ, n 1! n k! which can be treated as H l σ n 1 σ n k σ 2 k by neglecting constant terms, where n i 3, k 1, and σz = zk + 2η 1 t3 X πz cosh, c as defined in 2.7 in Section 2.1. For the derivative of the above sum of products, we start from the derivatives of σt. For r 1, σ r z = 2η 1 t 3 X π c 2η 1 t 3 X π c r cosh πz r is even; r sinh πz r is odd Evaluate 3.23 at γ in 2.5 yields, σ r γ = η 1 X π r 2 X ck πc + 4π2 t 3 X c r r is even; 3.24 K π π r K r r is odd. Then, ck X 4 2π, which is equivalent as X c r K r implies σ r γ X c r. 3.25

43 36 Hence, each derivative of σ n i γ produces 1, as does each derivative of σ 2 γ k, and by the chain rule we get the additional factor Kc/ X. Specifically, since n 1 + n 2 + n k = 2ν l + 2k with each n i 3 as we defined in 2.4, d r 1 dt r 1 [σn 1 γ σ n k γ] X k 2 c k 2ν l+2k+r 1 r1 Kc X and d r 2 dt r 2 [σ2 γ k ] ck 2k r2 X k 2 Kc X r2, which implies d r dt r [ σ n 1 ] γ σ nk γ σ 2 γ k 1 r Kc 1 2ν l X 2ν l for r, r 1, r 2 0. So, under the assumptions K ε = K µ K 1 ε and s 0, the desired derivatives of the sum of products term in H 2ν γ defined in 2.4 is bounded: d s dt s Hl σ n 1 σ n k σ 2 k Kc/ X s, which implies d s Kc s dt H 2νγ K µs. s X Therefore, G s j t K µs. 3.26

44 37 follows: For the phase term θt, its first derivative with x = n 2 n 2 1/c 3 is computed as θ t = 3η 3 α βx β t 3β 1 + 3η 2 n2 n 2 1 c X + 3η 2 1 X Kc 2πc t + 4π 2 t X Suppose β = 1 3 and set θ t = 0, we obtain Kc t X 2 + 4π 2 t = 2 3 πcη 1 n η n α + 3η 1 2 X c Recall that we have ck X 4 2π and 1 t 2 1/3. Thus, the left-hand side of 3.28 is of the size O1. This means we will get stationary points if the long coefficient of X 1/6 on the right-hand side of 3.28 is very close to X 1/6. Thus, we can divide the case into three: case 1: α 3 c + n n X 1 6 ; case 2: α 3 c n n X 1 6 ; case 3: α 3 c + n n X 1 6. In other words, if we apply the weighted first derivative test stated in emma 2.3 to the integrals I η 1η 2 η 3 j t in 3.20, the integrals will be negligible for all cases except the above three. Recall the bound of derivatives of the θ term in 3.22, θ r t X/c for r 2. Also put the bound we obtained for the derivatives of G j term in 3.26

45 38 into consideration, we may choose T = X/c, M = 1, U = 1 and N = K µ in emma 2.3. Also check that when 1 t 2 1/3, M = 1 2 1/3 1. Then, three error terms showed up by applying the weighted first derivative test in emma 2.3 to I η 1η 2 η 3 j t are O +O K µ [n/2] j=1 n+1 n j c X t=j K µn j t K µ + 1K nµ c X n+1 n n+1 n j c +O K µn j t. X j=1 t=0 Therefore, I η 1η 2 η 3 j t c X. So, what we left to do is to deal with the three exceptional cases defined above. Recall that c 4 2X/K 1 ε/2 and θ t X/c are results we already obtained. Then, apply the second derivative test stated in emma 2.1 to I η 1η 2 η 3 j t, we have I η 1η 2 η 3 j t c1/2 X 1/4 1 K 1 ε/2, which is a better bound than the trivial one O1. Therefore, plug in all obtained

46 39 bounds back to 3.19 with x = n 2 n 2 1/c 3 yields n2 n 2 1 Ψ 0 c 3 r n2 n 2 1 j c 2 X ν j 3 j=1 +O c 3 n2 n 2 2 r 3 1 X c 3 In case 1: α 3 c + n n X 1 6, we have c X 1/6 and n 2 n 2 1 X 1/2. Then, n 2 n 2 1 c 3 X n 2 n 2 1X 1 2 X 1 2. So, the summation among j in 3.29 is convergent. By picking up the first term, we have n2 n 2 1 Ψ 0 c 3 n2 n c c 3 2 X ν c 3 2 X ν Then, plug in all obtained bounds back to R η 1 ν in 3.15 at the very beginning of section 3.2, we have R η 1 ν K 2ν+1 n 1 c, n 2 >0 n 2 n 2 1 X1/2 c 1 c 4 2X K 1 ɛ ν+ 1 2 A f n 2, n 1 S n 1 n 2 zmod c z,c=1 z, n 2 ; z e c c n 1 c 3 2 X ν

47 40 Note that the formulas 4.22 in i [17] and 6.18 in Mckee-Sun-Ye [20] states that 0 z<c z,c=1 z e S z, n 2 ; c c n 1 = = u mod cn 1 1 uū 1 mod cn 1 1 u mod cn 1 1 uū 1 mod cn 1 1 mc1+ε n 1. n2 ū S0, 1 + un 1 ; ce cn 1 1 n2 n 1 ū e c z mod c z,c=1 1 + un1 z e c Also note that by assuming K X, we have c 4 2X/K 1 ε/2 K ε/2. And by the classical result that n 1,n 2 A f n 2,n 1 n 2 n 2 1 s converges absolutely for s > 1, we have n 1 c, n 2 >0 n 2 n 2 1 X A f n 2, n 1 n 2 n 2 1 n 1,n 2 n 2 n 2 1 X X ɛ. A f n 2, n 1 n 2 n 2 1 Hence, R η 1 ν K 2ν+1 c ν+ 1 2 c K ε/2 c 3 2 X ν K 2ν+1 c ν+ 1 c K ε/2 n 1,n 2 n 2 n 2 1 X1/2 K 2ν+1 X ν K ε 1+ν+ε K 1+ε X 1 2 +ε 2 K ε X ν 2 c 1+ε c 3 2 X ν A f n 2, n 1 c 1+ε n 2 n 2 1

48 41 Under the assumption that K X, ν R η 1 ν K 1+ε X 1 2 +ε. Combine with the bound for the continuous spectrum term 3.8 that we have obtained in section 2.1, which is K ε X 1+ε, we have K k j K+ S X f g j, α, 1 3 K1+ε X 1 2 +ε + K ε X 1+ε. For case 2: α 3 c n n X 1 6 and case 3: α 3 c + n n X 1 6, we have c 3n n α + O X 1/6, which implies n n α c 3 + OX1/6. Thus, if c X 1/6 and n 1/3 2 n 2/3 1 X 1/6, we go back to case 1. Otherwise, for c X 1/6, we have n n c α 3 + O X c Thus, X n 2n 2 1 c 3 X. So, the summation among j in 3.29 is convergent for both cases and we can only

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