| Co-Investigator(Kenkyū-buntansha) |
HIKARI Michitaka Keio Univ.Dept.Econ.Professor, 経済学部, 教授 (30056296)
NISHIOKA Kumiko Keio Univ.Dept.Econ, Professor, 経済学部, 教授 (80144632)
SHIOKAWA Iekata Keio Univ.Dept.Math, Professor, 理工学部, 教授 (00015835)
TOSE Nobayuki Keio Univ.Dept.Econ, Professor, 経済学部, 教授 (00183492)
WATABE Mutsuo Keio Univ.Dept.Buisiness & Commerse Professor, 商学部, 教授 (30080493)
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| Research Abstract |
1. Studies on various behaviours of multiple zeta-functions
Our 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 series
In 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
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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-functions
Regarding 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
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