文档介绍：11
Hardy-Ramanujan Journal
(2001) 11-19
On the values of the Riemann zeta-function
at rational arguments
S. Kanemitsu, Y. Tanigawa and M. Yoshimoto
Dedicated to Professor Takashi Yanagawa birthday with great respect
Abstract
In our previous papers [3], [4] we obtained a closed form evaluation of Ramanujan’s type
of the values of the (multiple) Hurwitz zeta-function at rational arguments (with denominator
even and numerator odd), which was in turn a vast generalization of D. Klusch’s and M.
Katsurada’s generalization of Ramanujan’s formula. In this paper we shall continue our
pursuit, specializing to the Riemann zeta-function, and obtain a closed form evaluation
thereof at all rational arguments, with no restriction to the form of the rationals, in the
critical strip. This is plete generalization of the results of the aforementioned two
authors. We shall obtain as a byproduct some curious identities among the Riemann zeta-
values.
1 Introduction and notation
In this paper we shall give a closed form evaluation of Ramanujan’s type of the values
of the Riemann zeta-function at positive rational arguments in the critical strip.
We have launched on this evaluation problem of zeta-values at rational arguments in
[2], where we have eeded in getting a Ramanujan type formula for ζ(2/3) after examining
the results of D. Klusch [6] and M. Katsurada [5].