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（评估ΔIId2aye d2（x））与对riemann zeta-functionζ（s）的均方根公式中的剩余项eiid2σyed2（t）的评估之间存在很强的比喻。研究它们的基本工具分别是Voronoi的公式和Atkinson的公式。我们在此主题上获得的结果是：1。通过使用Voronoi和Atkinson的公式，我们在短间隔中获得了ΔID2ayeD2（X）或EAD2σYeD2（t）差异的均方根公式。同样，我们在ζeea d12iye d1（s）.2的近似功能方程中为剩余项提供了相似的结果。我们已经研究了对字符的案例的概括。3。我们展示了Rankin-Selberg L功能核心的Riesz总和的Voronoi-Type公式，并证明了其均方根公式4。我们开发了使用Mellin-Barnes类型积分的方法，并确定了该方法在研究分析延续和不对称扩展研究中的实用性。此外，我们还可以证明LERCH ZETA函数的关节宇宙，以及与模块化形式相连的L型函数的宇宙。

项目成果

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。

I.Kiuchi,Y.Tanigawa: "The mean value theorem of the Riemann zeta-function in the critical strip for short intervals""Number Theory and its Applications" Kluwer. 231-240 (1999)

I.Kiuchi，Y.Tanikawa：“短区间临界带中黎曼 zeta 函数的中值定理”《数论及其应用》Kluwer。

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。

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。

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。