Behaviours of multiple Zeta-functions

1. 多项ζ函数的各种行为研究。拟完成多项级数S（μ, υ, α, β; μ, υ, ω）解析延拓的初步研究（已在项目申请表中介绍）。发现S （μ, υ, α, β; μ, υ, ω）的扭转部分S^^ ~（在应用表格中提到）具有涉及合流超几何函数Ψ （a, c; z）的无穷级数表示。S （μ, υ, α, β; μ, υ, ω）的这些初步结果使我们能够将其渐近行为视为α， β→+∞，以及它在某些整数格点上的特殊值。(1)对theta型和lambert型级数的某些公式的推广在他著名的笔记中，Ramanujan提出了theta型级数的完全渐近展开式的存在[]SU.[]，然后由Berndt和Evans计算了它的精确形式。在同一本笔记本中，他还发现了lambert型se的变换公式…More ries (]SU.[)，而κ是一个任意整数。本项目的首席研究员通过应用巴恩斯多重ζ函数的泛函性质，成功地推广了这两个公式。所得结果整理在论文《关于某theta型级数的Ramanujan的渐近公式》（将发表在某研究期刊上）和《关于奇数处黎曼ζ函数的特定值的Ramanujan的公式》（准备提交）中。(2) Lerch ζ函数的均值对于申请书中提到的Lerch ζ函数φ(λ, α, s)，首席研究员成功地得到了其均值（]SU.[）和（]SU.[）表示对变量s的第k阶导数的完全渐近展开式ims→±∞。所得结果整理在《melin - barnes型积分在Lerch ζ函数均方中的应用II》和《III》中。这两篇论文都将在一些学术期刊上发表。少

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