Journal of the Ramanujan Mathematical Society

Nobushige Kurokawa ¹, Hiroyuki Ochiai ², Masato Wakayama ³

Affiliations
1 Department of Mathematics, Tokyo Institute of Technology, Oh-Okayama Meguro, Tokyo-152-0033, JP
2 Department of Mathematics, Nagoya University, Chikusa, Nagoya-464-8602, JP
3 Kyushu University, Hakozaki Fukuoka-812-8581, JP

Abstract

We study basic properties of Milnor’s multiple gamma functions. Moreover, we introduce new multiple sine functions from these Milnor’s functions and show the difficult special values of the Dirichlet L-functions can be expressed by these functions.

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Nobushige Kurokawa ¹, Hiroyuki Ochiai ¹, Masato Wakayama ²

Affiliations
1 Department of Mathematics, Tokyo Institute of Technology, Meguro, Tokyo 152-8551, JP
2 Graduate School of Mathematics, Kyushu University, Fukuoka 812-8581, JP

Abstract

The purpose of this paper is to express the value of the Riemann zeta function at the positive odd integers in terms of the multiple sine functions. We also study the multiple trigonometric functions including the multiple cosine functions. Especially, we derive the algebraic differential equation satisfied by the multiple trigonometric functions.

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Nobushige Kurokawa ¹, Eva-Marie Muller-Stuler ¹, Hiroyuki Ochiai ¹, Masato Wakayama ²

Affiliations
1 Department of Mathematics, Tokyo Institute of Technology, Meguro, Tokyo-152-8551, JP
2 Graduate School of Mathematics, Kyushu University, Fukuoka-812-8581, JP

Abstract

We present the notion of the sine function of a ring, and we calculate some examples. This contains the case of the rational integer ring and the case of the imaginary quadratic integer rings. These results indicate such a sine function of a ring would be a good candidate to the aforementioned special function for the Kronecker’s Jugendtraum.

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Nobushige Kurokawa ¹, Masato Wakayama ²

Affiliations
1 Department of Mathematics, Tokyo Institute of Technology, Meguro Tokyo, 152-0033, JP
2 Faculty of Mathematics, Kyushu University, Hakozaki, Fukuoka, 812-8581, JP

Abstract

We prove a certain expression for ζ(3) using a theory of multiple sine functions. This expression has an interesting character indicating a “regulator” part as log S₃1/2, S₃(x) being the triple sine function. The value S₃(1/2) is interpreted also as the Mahler measure or the entropy of the associated dynamical system.

Keywords

Riemann Zeta, Triple Sine Function, Multiple Sine Functions, Selberg Zeta.

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