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Behaviours of multiple Zeta-functions

多个 Zeta 函数的行为

基本信息

批准号：
11640038
负责人：
KATSURADA Masanori
金额：
$ 2.3万
依托单位：
Keio Universtiy
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2000
项目状态：
已结题

项目摘要

1. Studies on various behaviours of multiple zeta-functionsOur primary study on the analytic continuation of the multiple series S(μ, υ, α, β ; μ, υ, ω) (introduced in the project application form) are going to be completed. It was found that the torsion part S^^〜 (mentioned in the application form) of S (μ, υ, α, β ; μ, υ, ω) has an infinite series representation involving the confluent hypergeometric function Ψ (a, c ; z). These preliminary results for S (μ, υ, α, β ; μ, υ, ω) enable us to treat its asymptotic behaviours as α, β→+∞, as well as its particular values at certain integer lattice points.2. Applications of multiple zeta-functions(1) Generarizations of certain formulae for theta-type and Lambert-type seriesIn his celebrated notebook, Ramanujan suggested the existence of a complete asymptotic expansion for the theta-type series (]SU.[) and then its exact form was computed by Berndt and Evans. In the same notebook he also found a transformation formula for the Lambert-type se … More ries (]SU.[) and κis an arbitrary integer. The head investigator of this project succeeded in generalizing these two formulae by applying the functional properties of Barnes' multiple zeta-function. The results obtained are arranged in the papers " On an asymptotic formula of Ramanujan for a certain theta-type series" (to appear in some research periodical), and " On a formula of Ramanujan for specific values of the Riemann zeta-function at odd integers" (prepared for submission).(2) Mean values of Lerch zeta-functionsRegarding the Lerch zeta-functions φ(λ, α, s) mentioned in the application form, the head investigater succeeded in obtaining the complete asymptotic expansions as Im s→±∞ for the mean values (]SU.[) and (]SU.[) denotes the κ-th derivative with respect to the variable s. The results obtained are arranged in the papers "An application of Mellin-Barnes type of integrals to the mean square of Lerch zeta-functions II" and "III", both of which will be submitted for publication in some academic journals. Less

1.多重zeta函数的各种性质的研究我们对多重级数S（μ，λ，α，β ; μ，λ，ω）的解析延拓（在项目申请表中介绍）的初步研究即将完成。发现S（μ，ε，α，β ; μ，ε，ω）的挠率部分S^^ε（在应用形式中提到）具有包含合流超几何函数ε（a，c ; z）的无穷级数表示.这些初步结果使我们能够处理S（μ，β，α，β ; μ，β，ω）的渐近行为为α，β→+∞，以及它在某些整数格点的特殊值.多重zeta函数的应用（1）θ型和Lambert型级数某些公式的推广在Ramanujan著名的笔记本中，他提出了θ型级数（SU）的完全渐近展开式的存在性。[1]然后它的精确形式由Berndt和Evans计算出来。在同一个笔记本中，他还发现了Lambert型se的变换公式。 ...更多信息 ries（）SU. [）且κ是任意整数。该项目的首席研究员成功地推广了这两个公式，通过应用巴恩斯的多重zeta函数的功能特性。所获得的结果被安排在文件“在一个渐近公式的拉马努金为一定的θ型系列”（出现在一些研究期刊），和“在一个公式的拉马努金为特定值的黎曼zeta函数在奇数”（准备提交）。(2)关于申请表中提到的Lerch zeta函数φ（λ，α，s），本文的主要研究者成功地得到了其平均值（]SU）在Im s→±∞时的完全渐近展开式。[]和（）。[）表示关于变量s的k阶导数。所得到的结果被安排在论文“梅林-巴恩斯型积分的应用，以均方的Lerch zeta函数II”和“III”，这两个都将提交发表在一些学术期刊。少