On the values of the Riemann zeta-function at rational arguments

Kanemitsu S., Tanigawa Y., Yoshimoto M.

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 a complete generalization of the results of the aforementioned two authors. We shall obtain as a byproduct some curious identities among the Riemann zetavalues.