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Studies on multiple zeta values, Arakawa-Kaneko zeta functions and iterated log-sine integrals

UMEZAWA, RYOTA 梅澤, 瞭太 名古屋大学

2020.04.02

概要

In this thesis, we discuss relations between multiple zeta values, Arakawa-Kaneko zeta functions and iterated log-sine integrals. In Section 1, we describe the background and known results of multiple zeta values, Arakawa-Kaneko zeta functions and log-sine integrals. In Section 2, we discuss analogues of the Arakawa-Kaneko zeta function based on [40]. In particular, we discuss an analogue of the ArakawaKaneko zeta function of Miyagawa-type and obtain relations among Miyagawa multiple zeta values. We also discuss a class of multiple zeta functions to which Ito’s zeta functions of the case of general index are related, and we obtain relations among special values of multiple zeta functions in this class. In Section 3, we define iterated log-sine integrals and prove its fundamental properties and discuss relations between iterated log-sine integrals, multiple zeta values and multiple polylogarithms. In particular, we give a method to obtain relations among multiple zeta values, which uses iterated log-sine integrals. We based on [41] for the method and the definition and fundamental properties of iterated log-sine integrals, and based on [42] for the rest of Section 3. We also prove theorems that iterated log-sine integral can be written in terms of multiple zeta values and multiple polylogarithms. These theorems make numerical evaluations of iterated log-sine integrals possible. Finally, we state some conjectures on multiple zeta values, multiple Clausen values, multiple Glaisher values and iterated log-sine integrals suggested by numerical evaluations.

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