Behaviours of multiple Zeta-functions

1.多重zeta函数的各种行为的研究我们即将完成对多重系列S(μ，υ，α，β；μ，υ，ω)(在项目申请表中介绍)的解析延拓的初步研究。发现S(μ，υ，α，β；μ，υ，ω)的扭转部分S(申请表中提到的)有一个涉及汇合超几何函数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它的渐近行为视为α，β→+∞，以及它在某些整数格点上的特定值。多重Zeta函数的应用(1)theta型和Lambert型级数某些公式的推广在他著名的笔记本中，Ramanujan提出了theta型级数()SU[)的完全渐近展开式的存在，然后由Berndt和Evans计算了它的确切形式。在同一本笔记本中，他还发现了Lambert型Se…的变换公式More Ries(]Su.[)和κ是任意整数。这一项目的首席研究员通过应用巴恩斯多重Zeta函数的函数性质成功地推广了这两个公式。所得结果列在《关于某一theta型级数的Ramanujan渐近公式》(发表在某些研究期刊上)和《关于Riemann Zeta函数在奇整数处的特定值的Ramanujan公式》(准备提交)。(2)申请表中提到的Lerch Zeta函数的平均值(关于Lerch Zeta函数φ(λ，α，S)，主要研究人员成功地得到了如下的完全渐近展开式：Im S→±∞的平均值()SU[)和(]SU[]表示关于变量S的κ阶导数。所得结果被安排在论文《Mellin-Barnes型积分对Lerch Zeta函数均方的应用II》和《III》两篇论文中，这两篇论文将提交一些学术期刊发表。较少

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