On The Estimates For Exponential Sums Over Primes Containing Fourier Coefficients Of Cusp Forms

In prime number theory for an any given sequence an one of basic and impor-tant question is to investigate cancellations in the exponential sum containing these numbers.As a hot subject in modern number theory any strides been made on the distribution of the Fourier coefficients of modular forms will further increase the understandings of modular functions for people.Generally one would like to study the estimates for exponential sums over primes containing Fourier coefficients of cusp forms.Let/be a primitive holomorphic or Maass cusp form for SL(2,Z),and denote by af(n)the nth normalized Fourier coefficient of it.In this thesis we shall estimate the following sumwhere α ∈ R,Δ(n)is the von Mangoldt function andt:R →C is an arithmetic function.Recently Fouvry and Ganguly[8]investigated the resonance behavior of the Fourier coefficient af(n)in the exponential sums over primes.They showed that there exists an effective constant c>0 such that for any α ∈Rwhere the implied constant depends on the form f.In Chapter 1 we shall study the oscillations of Fourier coefficients of cusp forms in quadratic exponential sum.Theorem 1 Let f be a primitive holomorphic or Maass cusp form for S L(2,Z).Let af(n)denote the nth normalized Fourier coefficient of the form f.Let N ≥ 2.Then there exists an effective absolute c>0 such that,for any α,β∈ R,there exists an effective constant C(f)>0 such thatAs an application of Theorem 1 we are concerned with the quadratic Waring-Goldbach problem.Loo-Keng Hua[17]showed that every sufficiently large num-ber N congruent to 5 modulo 24 can be represented as five squares of primes,and the number of such representations is O(N3/2/log5 N).More generally for any positive number s ≥ 5 we have Here C = C(s)>0 is an absolute constant and(?)s(N)is the singular series.In Chapter 1 we are able to prove a better bound compared with the error term in the asymptotic formula above.The improvement essentially stems form the oscillations of the argument of coefficients AF(p,1).Corollary l Let the form f and the coefficient af(n)be as in Theorem 1.Then there exists an effective positive c such that,for every N ≥ 4 and every s ≥ 5 uniformly we have where the implied constant depends only on the form f.In Chapter 2 we will investigate the linear exponential sum containing the Fourier coefficients of Hecke-Maass forms for S L(3,Z).We are devoted to study-ing the Prime Number Theorem for the coefficients of L(s,F)with twists(by ad-ditive characters e(a)= e2πiα).So far there is no any progress been made at prime arguments which is absolutely play a key ro1e in the study of modular forms.As an attempt to apply the theory of automorphic forms to classical number theory we shall firstly give an estimate which is of independent of the parameter α.Theorem 2 Let F be a Hecke-Maass form for SL(3,Z).Let AF(n,1)denote the(n,l)-th normalized Fourier coefficient of the form F.Let N ≥ 2.Then there exists an effective absolute c>0 such that,for any α∈R,there exists an effective constant C(f)>0 such thatAs an application of Theorem 2 we are concerned with the Vinogradow’s three primes theorem.We are able to prove the following corollary.Corollary 2 Let the form F and the coefficient AF(n,1)be as in Theorem 2.Then there exists an effective constant c>0 such thatfor every N ≥4,for every sequence of complex numbers(αa)a≥1 and(βb)b≥1,whereand the implied constant depends only on the form F.In particular we obtain an analogue of Vinogradov’s three primes theorem[49]associated to the coefficient AF(n,1).We have either of the following sumsandis O(N2 exp((?))),for some constant c>0,where the implied O-constant depends only on the form F.Another main subject in this thesis is to study the non-linear exponential sums involving the Fourier coefficients of cusp forms.In Chapter 3 we will proceed our argument on these problems involving the Fourier coefficients of the S L(2,Z)cusp forms and their symmetric-square lifts over primes.This part of work generalizes the works of Zhao’s[52]and Pi and Sun’s[35].Theorem 3 Let f be a primitive holomorphic or Maass cusp form for S L(2,Z).Let af(n)denote the nth normalized Fourier coefficient of the form f.Let ≥ 2.Then for any 0<α ≤ 1/2 and any η ≠0 we have for some effective c>0,where the implied constant depends on the form f,η and a.Assuming the Grand Riemann Hypothesis(GRH for short)for L(f,s),the upper bound can be replaced by N(1+α)/2.’Theorem 4 Let the form f and the coefficient af(n)be as in Theorem 3.Suppose that F is the symmetric-square lift of the form f,with AF(n,1)denoting the n-th coefficient of the L-function for it Then for any 0<α<(3 +(?))/8 and any η≠ 0 we have where the implied constant depends on F,η and ε.As an important arithmetic function in number theory the study of the Mobious function μ(n)is of great interest.Recently Pi and Sun[36]showed that for any η ≠ 0 where the implied constant depends on the form f and η.In Chapter 3 we will establish the following theorem which reveals some new features.Theorem 5 Let the formf and the coefficient af(n)be as in Theorem 3.Let N ≥2.Then for any 0<α ≤ 1/2 and any η≠0 we have for some effective c>0,where the implied constant depends on the form f,77 andα.Moreover suppose that F is the symmetric-square lift of the form f.Then for any 0<α<4/9,any η≠0 and any ε>0 we havefor some effective c>0,where the implied constant depends on F,η and ε.