Behaviours of multiple Zeta-functions

点击翻译按钮获取中文摘要

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