【作者】 张伟；

【导师】 任秀敏；

【作者基本信息】 山东大学 ， 基础数学， 2020， 博士

【摘要】 设f是关于SL2(Z)的一个全纯Hecke尖形式或Hecke-Maass尖形式,则f有Fourier展开.设λf(n)为其正规化的Fourier系数,定义Hecke L-函数L(s,f)L(s,f)=(?)=(?)(1-λf(p)p-s+p-2s)-1,R(s)>1.众所周知在,L(s,f)具有与经典的Dirichlet L-函数L(s,x)类似的性质与函数方程.例如,L(s,f)可以解析延拓到整个复平面,并且其非显然零点均介于0<rs<1之间.关于L(s,f)的广义黎曼猜想是说L(s,f)的所有非平凡零点均落在直线Rt(s)=1/2上.而零点密度猜想是说Nf(a,T)《T2(1-σ)logc T或者不带logT的形式Nf(σ,T)《T2(1-σ)+e,这里c>0,并且ε>0为一个任意小的数.其中Nf(σ,T)表示Hecke L-函数L(s,f)在区域{σ≤β≤1，|γ| ≤T}中的零点个数.当f是关于SL2(Z)的一个全纯Hecke尖形式时,Ivic考虑了这个问题并且在[19]中证明了Nf(σ,T)《T(σ)(1-σ)+ε,其中(?)通过应用Karatsuba以及Voronin的方法,Yashiro在[50]中给出了上述结果的另一种证明.在2007年,Sankaranarayanan以及Sengupta[45]研究了当f是关于SL2(Z)的Hecke-Maass尖形式的情形,并且证明了Nf(σ,-T)<<T4/(1-σ)/3-2σlog26T,σ≥1/2+1/logT.2011年,Xu[48]在3/4<σ<1改进了上述结果,证明了Nf(σ,T)《T(8σ-5)(1-σ)/-2σ2+6σ-3log57T,3/4≤σ<1.2013年Tang[47]做出了进一步的改进,证明了存在一个常数c>0使得Nf(σ,T)《T2/σ(1-σσ)logcT,3/4≤σ<1.如果不考虑Tε与logcT的差别,这一结果与Ivic在全纯Hecke尖形式下的结果是一致的.在本文中,我们将证明以下定理.这一结果分别改进了 Ivic和Tang在3/4<σ<1上的结果.定理1 设f(z)是关于SL2(Z)的全纯Hecke尖形式或Hecke-Maass尖形式.那么对于Hecke L-函数L(s,f),我们有Nf(σ,T)《TA(σ)(1-σ)+ε,其中并且隐含的大O常数依赖于f以及ε.我们知道,经典Dirichlet L-函数的零点密度估计在许多重要的数论问题中有着重要的应用,特别是在涉及关于素数分布问题的研究中其常常可以替代广义黎曼猜想并给出很强的结果,比如小区间上的素数定理,孪生素数分布,素变数指数和的估计等等.利用Nf(σ,T)的估计,可以考虑与自守形式f有关的类似问题.设f是关于SL2(Z)的全纯Hecke尖形式,Af(n)为其正规化的Fourier系数,本文将研究如下素变数指数和S(α,θ,x)=λf(p)(logp)e(αpθ),其中α≠0且0<θ<1.这个问题被许多专家学者所研究,例如对于θ=1/2,固定的 α=-2,Iwaniec,Luo 以及 Sarnak(参考[25])给出了一个具有启发意义的条件性结果S(α,1/2,x)=cfx3/4+O(x8+ε),其中cf是一个只依赖于尖形式f的常数.对固定的α>0,Zhao[54]在2006年证明了S(α,1/2,x)《x5/6(log x)21,其中隐含的大O常数依赖于α以及尖形式f.2010年,对于一般的θ∈(0,1),在[42]中,Pi以及Sun考虑了这个问题.对于α≠0以及θ∈(0,9/32]∪[1/2,9/16],他们证明了S(α,θ,x)《xε(x1+θ/2+x3/4+θ/6+x1-θ/2),其中隐含的大O常数依赖于α,θ,ε以及尖形式f.应用Vaughan恒等式,对于α≠0,以及0<θ≤1/2,Hou[12]证明了存在绝对常数c>0使得S(α,θ,x)≤(x5/6+x1-θ/2)(lox)c,其中隐含的常数依赖于α,θ,c以及尖形式f.最近,对于α>0以及0<θ<1,Jiang以及Lii[26]进一步改进了上述结果,证明了如下估计当0<θ<1,当3/4<θ<1,其中隐含的大O常数依赖于α,θ以及尖形式f.应用[44]以及[42]中的一些方法,在本文中,我们将证明下面的估计.定理2 设α≠0为固定的实数并且θ满足0<θ<1.那么存在一个常数c≥ 28使得对于充分大的x,我们有S(α,θ,x)《(x3/4+θ/6+x9+5θ/14+x1-θ/2)logcx,其中隐含的大O常数依赖于α,θ,c以及尖形式f.很容易看出在1/3<θ<1/2以及1/2<θ≤ 3/4上,这个估计要比之前的结果好.另外当0<θ ≤ 9/16时,我们有S(α,θ.x)《(x§+§+x1-§)logcx.这一结果在某种程度上改进了之前的结果并且给出了S(α,θ,x)在0<θ<9/16上的一致性估计.在[42]中,Pi以及Sun还考虑了如下的指数和问题λf(p)(logp)e(kαpθ).对于任意的α≠0,θ ∈(0,9/32]和 kk ∈[1,x9/32-θ]或 θ ∈[1/2,9/16]以及k∈[1,X9/16-θ],他们得到λf(p)(logp)e(kαpθ)《(k1/2x1/2+θ/2+k1/6x3/4+θ/6+k-1/2x1-θ/2)xε,其中隐含的大O常数依赖于α,θ,ε以及尖形式f.注意到上面的结果没有给出θ∈(9/32,1/2)以及θ ∈(9/16,1)的估计.作为补充,我们将证明如下定理.定理 3 设 α≠0,对 θ ∈(0,9/16]以及 k∈[1,X9/16-θ],我们有λf(p)(logp)e(kαpθ)《k1/2x1/2+θ/2+k1/6x3/4+θ/6+k-1/2x1-θ/2)xε,其中隐含的大O常数依赖于α,θ,ε以及尖形式f.此外,对 θ ∈(9/16,1)以及 k∈[x9/16-θ,x1-θ],我们有λf(p)(logp)e(kαpθ)《kη1(k,x,θ)xη2(k,x,θ)+k1/2x1+θ)(logx)28,其中η1(k,x,θ)=3/2-η2(k,x,θ)=3/2+3θ/2-并且隐含的大O常数依赖于α,θ以及尖形式f.受到上述问题的启发,很多人也研究了关于Hecke-Maass尖形式的素变数指数和估计问题,即U(α,θ,x)=λg(p)(logp)e(αpθ),其中S是关于S/2(Z)的一个Hecke-Maass尖形式,λg(n)为其正规化的Fourier 系数.对于α≠o以及0<θ≤ 1/2,Hou[12]证明了存在常数c>0使得U(α,θ,x)《(x5/6+ x1-θ/2)(log x)c,其中隐含的大O常数依赖于α,θ,c以及尖形式g.对于α>0,Jiang以及Lu[26]得到U(α,θ,x)《(X5/6+x2+θ/3+x1-θ/2)(log x)9/2 当 0<θ<1,其中隐含的大O常数依赖于α,θ以及尖形式g.对于这类指数和,我们将证明下面的估计.定理4 设α≠0为固定的实数并且θ满足0<θ<1-7/64=0.890625.那么存在一个常数c ≥ 28使得对于充分大的x,我们有U(α,θ,x)《(x3/4+θ/6+x9+5θ/14+x1-θ/2)logc x+x7/64+θlogx其中隐含的大O常数依赖于α,θ,c以及尖形式g.注意到当0<θ ≤ 1-21/128时,有7/64+θ ≤(2+θ)/3.因此很容易看出在 1/3<θ<1/2 以及 1/2<θ<107/128=0.8359375 上,定理 4 要比之前的结果好.根据定理4以及定理4的证明,我们有下面的推论.推论1 假设L-函数/(s,g)的零点密度估计满足Ng(σ,T)《ε,gT2(1-σ)+ε.设α≠0为固定的实数并且θ满足0<θ<1-7/64=0.890625.那么存在一个常数c≥ 28使得对于充分大的x,有λ9(p)(logp)e(αpθ)《x7/64+θlogx+(x2-θ/2+x1+θ/2)xε,其中隐含的大O常数依赖于α.θ,ε以及尖形式g.推论2 假设I-函数/(s,g)的零点密度估计满足Ng(σ,T)《ε,g T2(1-σ)+ε.那么对充分大的x,我们有λg(p)(logp)e(αp1/2)《x3/4+ε,其中隐含的大O常数依赖于α.θ,ε以及尖形式g.更多还原

【Abstract】 Let f be a holomorphic Hecke cusp form or Hecke-Maass cusp form on SL2(Z),then f has a Fourier expansion.Let λf(n)be its normalized Fourier coefficients and define the Hecke L-function L(s,f)L(s,f)=(?)=(?)(1-λ(p)p-s+p-2s)-1,R(s)>1.It is well known that L(s,f)has similar analytic properties and function equa-tions as the classical Dirichlet L-function L(s,χ).For example,L(s,f)can be continued analytically to the complex plane and its non-trivial zeros are in the critical strip 0<Rs<1.The GRH(Generalized Riemann Hypothesis)for L(s,f)states that all its nontrivial zeros are on the line R(s)=1/2.While the Zero-Density Hypothesis states that Nf(σ,T)《T2(1-σ)logc T or the form without log T Nf((σ,T)《T2(1-σ)+ε.Here c>0,and ε>0 is a sufficiently small constant.Let Nj(σ，T)denote the number of zeros of Hecke L-function L(s,f)in the region {σ≤β≤1，|γ| ≤T}When f is a holomorphic Hecke cusp form for SL2(Z),Ivic considered this problem in[19]and proved that Nf(σ,T)<<TA(σ)(1-σ)+ε,wherCombining the method of Karatsuba and Voronin,Yashiro gave another proof for the above results in[50].In 2007,Sankaranarayanan and Sengupta[45]investigated this problem when f is a Maass cusp form for SL2(Z),and proved the estimate Nf(σ,T)<<T4(1-σ)/3-2σlog26 T for σ≥1/2+1/logT.Later,Xu[48]improved this result when 3/4<σ<1 by showing that Nf(σ,T)<<T(8σ-5)(1-σ)/-2σ2+6σ-3 log57T for 3/4≤σ<1.Further improvement was proved by Tang[47]who showed that there exists a fixed constant c>0 such that Nf(σ,T)《T2/σ(1-σ)logc T for 3/4≤ σ<1.Tang’s result was obtained by using the method of Huxley and following the argument of Jutila[27]and Heath-Brown[9].From these estimates one can see that the best uniform bound for Nf(σ,T)in the range 1/2 ≤σ ≤1 is Ts(1-σ)/3 logc T.If the difference of Tε and logcT is ignored,this result is of the same quality with the result obtained by Ivic in the case of holomorphic Hecke cusp form.In this paper,we will prove the following.This improves the results of Ivic and Tang when 3/4<σ<1,respectively.Theorem 1 Let f(z)be a normalized holomorphic Hecke cusp form or Hecke-Maass cusp form for SL2(Z).Then we have Nf(σ,T)<<TA(σ)(1-σ)+ε,where#12 and the implied constant depends only on f and ε.We know that the zero density results of classical Dirichlet L-functions have important applications in many important number theory problems.In some problems,it can be used to replace the GRH and give very strong results,especially in the problems concerning the distributions of prime numbers.For example,the prime number theorem in short intervals,the distribution of twin prime numbers,exponential sum over primes and so on.By using the estimates of Nf(σ,T),we can consider similar problems related to the cusp form f.In this paper,we are interested in the following exponential sum S(a,θ,x)=(?)λf(p)(logp)e(αpθ),x<p<2x where α≠0 and 0<θ<1.This problem has been investigated by many authors.When θ=1/2,for fixed α=-2,Iwaniec,Luo and Saxnak(see[25])gave conditional heuristics result such that S(α,1/2,x)=cfx3/4+O(x5/8+ε),where cf is a constant depending only on f.In 2006,Zhao[54]showed that for α>0,S(a,1/2,x)《x5/6(log x)21,where the implied constants depend on α and the cusp form f.This is the first unconditional result for such type.However,there is no similar result for α<0 in[54].In 2010,Pi and Sun considered this problem for generalθ ∈(0,1)in[42]and proved that for α≠ 0 and θ∈(0,9/32]∪[1/2,9/16],there holds the estimate S(α,e,x)《xε(x1+θ/2+x3/4+θ/6+x1-θ/2),where the implied constant depends on α,θ,ε and the cusp form f.By using Vaughan’s identity,Hou[12]proved that for α≠0 and 0<θ≤1/2,there exists an absolute constant c>0 such that S(α,θ,x)《(x5/6+x1-θ/2)(logx)c,where the implied constant depends on α,θ,c>0 and the cusp form f.Recently,for α>0 and 0<θ<1,Jiang and Lu[26]improved this result by showing thatIn this paper,we borrow the idea in[44]and follow the argument in[42]to prove the following estimate.Theorem 2 Let α＝0 be fixed and 0<θ<1.Then there exists an absolute constant c ≥ 28 such that for x sufficiently large in terms of α,θ,there holds S(α,θ,x)《(x3/4+θ/6+x9+5θ/14+x1-θ/2)logcx,where the implied constant depends on α,θ,c and f.It is easy to see that this estimate is better than previous results when 1/3<θ<1/2 and 1/2<θ<3/4.Moreover,when 0<θ≤9/16,we have S(α,θ,x)《(x3/4+θ/6+x1-θ/2)logcx.This refines earlier results and gives a bound depending continuously on 0.In[42],Pi and Sun also consider the following exponential sumλf(p)(log p)e(hαpθ).For any α≠0,θ∈(0,9/32]and h ∈[1,x9/32-θ]or θ ∈[1/2,9/16]and h ∈[1,x9/16-θ],they obtained the estimate λf(p)(logp)e(hapθ)《(h1/2x1/2+θ/2+h1/6x3/4+θ/6+h-1/2x1-θ/2)xε,where the implied constant depends on α,θ,ε and the cusp form f.One may note that the above result does not cover the range of θ∈(9/32,1/2)and θ ∈(9/16,1).For complements,we will prove the following theorem.Theorem 3 For α≠0,θ ∈(0,9/16]and k ∈[1,x9/16∷θ we haveλf(p)(logp)e(kαpθ)《(k1/2x1/2+θ/2+k1/6x3/4+θ/6+k-1/2x1-θ/2)xε,where the implied constant depends on α,θ,ε and the cusp form f.Moreover,for θ∈(9/16,1)and k ∈[x9/16-θ,x1-θ],we haveλf(p)(log p)e(kαpθ)《(kη1(k,xθ)xη2(k,x,θ)+k1/2x1+θ/2)(log x)28,whereη1(k,x,θ)3/2-η2(k,x,θ)=3/2+3θ/2-and the implied constant depends on α,θ and the cusp form f.Inspired by the above results,some people also investigated the expo-nential sum over primes when g is a Hecke-Maass cusp form with respect to SL2(Z),i.e.U(α,θ,x)=λ(P)(log p)e(αpθ),where λg(n)are the normalized Fourier coefficients.For α≠0,and 0<θ≤1/2,Hou[12]obtained that there exists an absolute constant c such that U(α,θ,x)《(x5/6+x1-θ/2)(log x)c,where the implied constant depends on α,θ,c>0 and the cusp form g.For α>0,Jiang and Lu[26]obtained U(α,θ,x)《(x5/6+x2+θ/3+x1-θ/2)(log x)9/2 for 0<θ<1,where the implied constant depends on α,θ and the cusp form g.In this thesis,we will prove the following estimates.Theorem 4 Let α≠0 be fixed and 0<θ<1-7/64=0.890625.Then there exists an absolute constant c≥ 28 such that for sufficiently large x,we have U(α,.θ,x)《x3/4+θ/6+x9+5θ/14+x1-θ/2)logc x+x7/64+θlog x.where the implied constant depends on α,θ,c and the cusp form g.Note that 7/64+θ≤(2+θ)/3 for 0<0 ≤1-21/128.So it is easy to see that Theorem 4 is better than the previous results for 1/3<θ<1/2 and 1/2<θ≤107/128=0.8359375.By Theorem 4 and its proof,we can obtain the following corollaries.Corollary 1 Assume the zero-density estimates for L(s,g)satisfying Ng(σ,T)《g,εT2(1-σ+ε.For fixed α≠0 and 0<θ<1-7/64=0.890625 and sufficiently large x,we haveλg(p)e(αpθ)《x7/64+θlogx+(x2-θ/2+x1+θ/2)xε,where the implied constant depends on a,0,e and the cusp form g.Corollary 2 Assume the zero-density estimates for L(s,g)satisfying Ng(σ,T)《T2(1-σ)+ε.Then for fixed α≠0 and sufficiently large x,we haveλg(p)(logp)e(αp1/2)《x3/4+ε,where the implied constant depends on α,θ,ε and the cusp form g.更多还原