关于L-函数系数的若干问题

发布时间：2018-06-14 14:42

本文选题：自守形式 + 自守表示　；参考：《山东大学》2017年博士论文

【摘要】：L-函数是数论中神秘而特别常见的研究对象,最简单的例子就是Rie-mann ζ函数.类似于Riemann ζ函数,一般的L-函数也存在与之相关的广义Riemann假设、广义Ramanujan猜想等问题.众所周知,广义Ramanujan猜想在多数情况下仍是公开问题,在本文中,在不假设广义Ramanujan猜想成立的条件下,我们主要研究了 L-函数系数的一些分布规律.在第一章中,在不假设广义Ramanujan猜想成立的条件下,我们确立了一类L-函数系数的一般性求和公式.作为应用,我们考虑了 Hecke-Maass尖形式的Fourier系数的整次幂均值.在第二章中,我们集中研究了自守L-函数.设π是GLm(AQ)上酉自守尖点表示,以及L(s,π)是对应的π的自守L-函数,其在半平面Rs1上可表达成Dirichlet级数,即(?)(?)我们对λπ(n)的四次幂均值的上界非常感兴趣,即∑n≤x|λπ(n)|4.如果m = 2,我们考虑了λπ(n)的十六次幂均值.作为应用,我们研究了(?)的下界,改进了先前的对应结果.在第三章中,我们研究了关于SLm(Z)上Hecke-Maass尖形式的Bombieri-Vinogradov定理的类似形式.特别对SL2(Z)上全纯或Maass尖形式,SL2(Z)上全纯Hecke特征尖形式的对称平方提升以及Ramanujan猜想成立下SL3(Z)上的Maass尖形式,其对应的Fourier系数在素数点上的分布水平为1/2,我们得到SL2(Z)全纯尖形式或Maass尖形式的分布水平为1/2,这与经典的Bombieri-Vinogradov定理一样强.作为这些特殊情形下的应用,我们给出了一类转移卷积和在素数上的节余,即当α≠0,(?)(?)其中ρ(n)表示全纯尖形式f的Fourier系数λf(n),或其对称平方提升F的Fourier系数AF(n,1).进一步,作为结论,我们有渐进公式(?)(?)其中E1(α)是依赖于α的某个常数.
[Abstract]:The L- function is a mysterious and especially common object in the number theory. The simplest example is the Rie-mann zeta function. It is similar to the Riemann zeta function. The general L- function also exists in the generalized Riemann hypothesis and the generalized Ramanujan conjecture. As we all know, the generalized Ramanujan guess is still open in most cases, in this paper In the condition that the generalized Ramanujan conjecture is not assumed, we mainly study the distribution of the coefficients of the L- function. In the first chapter, we have established a general summation formula for a class of L- function coefficients without assuming that the generalized Ramanujan conjecture is established. As an application, we consider the Fouri of the Hecke-Maass sharp form. In the second chapter, we focus on the self defense L- function in the second chapter. Set Pi is the unitary point point representation on GLm (AQ), and L (s, PI) are the corresponding L- function of the corresponding PI, which can be expressed as Dirichlet series on the semi plane Rs1, that is, we are very interested in the upper bounds of the four power mean of lambda PI (n), that is, sigma n < x| [x|]. |4. if M = 2, we consider the sixteen power mean of lambda PI (n). As an application, we studied the lower bound of (?) and improved the previous corresponding results. In the third chapter, we studied a similar form of the Bombieri-Vinogradov theorem on the Hecke-Maass tip on SLm (Z), especially for SL2 (Z), holomorphic or Maass tip, SL2 (Z) holomorphic. The symmetric square lifting of Hecke characteristic cusp and the Maass sharp form on SL3 (Z) under the Ramanujan conjecture, the corresponding Fourier coefficient distribution at the prime number point is 1/2, we get SL2 (Z) Quan Chunjian form or Maass tip distribution level as 1/2, which is as strong as the canonical Bombieri-Vinogradov theorem. In different cases, we give a class of transfer convolution and the savings on the prime number, that is, when alpha 0, (?) (?) (?) in which p (n) represents the Fourier coefficient f (n) of holomorphic F, or the Fourier coefficient AF (n, 1) of its symmetric square lifting F. As a conclusion, we have a asymptotic formula (?) (?) in which E1 (a) is dependent on a constant of alpha.
【学位授予单位】：山东大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O156.4

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