本文選題：自守形式 + 自守表示��； 參考：《山東大學(xué)》2017年博士論文

【摘要】：L-函數(shù)是數(shù)論中神秘而特別常見的研究對象,最簡單的例子就是Rie-mann ζ函數(shù).類似于Riemann ζ函數(shù),一般的L-函數(shù)也存在與之相關(guān)的廣義Riemann假設(shè)、廣義Ramanujan猜想等問題.眾所周知,廣義Ramanujan猜想在多數(shù)情況下仍是公開問題,在本文中,在不假設(shè)廣義Ramanujan猜想成立的條件下,我們主要研究了 L-函數(shù)系數(shù)的一些分布規(guī)律.在第一章中,在不假設(shè)廣義Ramanujan猜想成立的條件下,我們確立了一類L-函數(shù)系數(shù)的一般性求和公式.作為應(yīng)用,我們考慮了 Hecke-Maass尖形式的Fourier系數(shù)的整次冪均值.在第二章中,我們集中研究了自守L-函數(shù).設(shè)π是GLm(AQ)上酉自守尖點表示,以及L(s,π)是對應(yīng)的π的自守L-函數(shù),其在半平面Rs1上可表達成Dirichlet級數(shù),即(?)(?)我們對λπ(n)的四次冪均值的上界非常感興趣,即∑n≤x|λπ(n)|4.如果m = 2,我們考慮了λπ(n)的十六次冪均值.作為應(yīng)用,我們研究了(?)的下界,改進了先前的對應(yīng)結(jié)果.在第三章中,我們研究了關(guān)于SLm(Z)上Hecke-Maass尖形式的Bombieri-Vinogradov定理的類似形式.特別對SL2(Z)上全純或Maass尖形式,SL2(Z)上全純Hecke特征尖形式的對稱平方提升以及Ramanujan猜想成立下SL3(Z)上的Maass尖形式,其對應(yīng)的Fourier系數(shù)在素數(shù)點上的分布水平為1/2,我們得到SL2(Z)全純尖形式或Maass尖形式的分布水平為1/2,這與經(jīng)典的Bombieri-Vinogradov定理一樣強.作為這些特殊情形下的應(yīng)用,我們給出了一類轉(zhuǎn)移卷積和在素數(shù)上的節(jié)余,即當α≠0,(?)(?)其中ρ(n)表示全純尖形式f的Fourier系數(shù)λf(n),或其對稱平方提升F的Fourier系數(shù)AF(n,1).進一步,作為結(jié)論,我們有漸進公式(?)(?)其中E1(α)是依賴于α的某個常數(shù).
[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.
【學(xué)位授予單位】：山東大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2017
【分類號】：O156.4

【參考文獻】

相關(guān)期刊論文 前1條

1 REN XiuMin;YE YangBo;;Resonance and rapid decay of exponential sums of Fourier coefficients of a Maass form for GL_m(Z)[J];Science China(Mathematics);2015年10期

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本文編號：2017745