Behavior of Zeta and L-functions and their arithmetic meaning

Zeta 和 L 函数的行为及其算术意义

• 批准号: 09440009
• 负责人: MATSUMOTO Kohji
• 金额: $ 7.42万
• 依托单位: Nagoya University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1999
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09440009/
• 关键词: Riemann zeta-function; Dirichlet L-function; Voronoi's formula; Divisor problem; Mean square; Asymptotic expansion; Rankin-Selberg L-function; Universality; Voronoiの公式; Lerchゼータ関数; ゼータ関数; L関数; 平均値定理; Pisot数; 二重ガンマ関数

There is a strong analogy between the divisor problem (the evaluation of ΔィイD2aィエD2 (X)) and the evaluation of the remainder term EィイD2σィエD2 (T) in the mean square formula for the Riemann zeta-function ζ (S). The basic tools for the study of them are Voronoi's formula and Atkinson's formula, respectively. The results we have obtained on this topic are :1. By using Voronoi's and Atkinson's formulas, we obtained the mean square formulas of the differences of ΔィイD2aィエD2 (X), or EィイD2σィエD2 (T), in short intervals. Also we proved similar results for the remainder term in the approximate functional equation for ζ ィイD12ィエD1(S).2. We have studied the generalization to the cases with characters.3. We showed the Voronoi-type formula for the Riesz sum of the coefficient of Rankin-Selberg L-functions, and proved their mean square formulas.4. We developed the method of using Mellin-Barnes type of integrals, and established the usefulness of this method for the study of analytic continuation and asymptotic expansions.Also, we could prove the joint universality for Lerch zeta-functions, and the universality of L-functions attached to modular forms.

除数问题(ΔィイD2aィエD2(X)的求值)与黎曼Zeta函数的均方公式中的余项EィイD2σィエD2(T)的求值有很强的相似性(S)。研究它们的基本工具分别是Voronoi公式和Atkinson公式。1.利用Voronoi公式和Atkinson公式，得到了ΔィイD2aィエD2(X)或EィイD2σィエD2(T)在短区间内的差的均方公式。对于ζィイD12ィエD_1(S)的近似函数方程中的余项，我们也证明了类似的结果。研究了它对具有特征的情形的推广。给出了Rankin-Selberg L函数系数的Riesz和的Voronoi型公式，并证明了它们的均方公式。我们发展了利用Mellin-Barnes型积分的方法，建立了该方法在研究解析延拓和渐近展开式方面的有效性，并证明了Lerch Zeta-函数的联合普适性和附加于模形式的L-函数的普适性。

项目成果

M.Katsurada: "An application of Mellin-Barnes type of integrals to the mean squore of L-functions" Liet.Mati.Rink.38. 98-112 (1998)

M.Katsurada：“Mellin-Barnes 型积分在 L 函数均方中的应用”Liet.Mati.Rink.38。

M.Katsurada: "Power series and asymptotic series associated with the Lerch zeta-function" Proc.Japan Acad.Ser.A. 74. 167-170 (1998)

M.Katsurada：“与 Lerch zeta 函数相关的幂级数和渐近级数”Proc.Japan Acad.Ser.A。

K.Matsumoto: "Asymptotic series for double zeta,double gamma,and Hecke L-functions" Math.Proc.Cambridge Phil.Soc.(to appear). (1998)

K.Matsumoto：“双 zeta、双 gamma 和 Hecke L 函数的渐近级数”Math.Proc.Cambridge Phil.Soc.（即将出现）。

F.Fujii and Y.Kitaoka: "On plain lattice points whose coordinates are reciprocals modulo a prime" Nagoya Math.J.147. 137-146 (1997)

F.Fujii 和 Y.Kitaoka：“在坐标为素数模倒数的普通格点上”Nagoya Math.J.147。

A. Laurineikas, K. Matsumoto: "A joint universality and the functional independence for Lerch zeta-functions"Nagoya Math. J.. (to appear).

A. Laurineikas、K. Matsumoto：“Lerch zeta 函数的联合普遍性和函数独立性”名古屋数学。