S Kanemitsu ; Y Tanigawa ; M Yoshimoto - On the values of the Riemann zeta-function at rational arguments

hrj:144 - Hardy-Ramanujan Journal, January 1, 2001, Volume 24 - https://doi.org/10.46298/hrj.2001.144
On the values of the Riemann zeta-function at rational arguments

Authors: S Kanemitsu ; Y Tanigawa ; M Yoshimoto

In a companion paper, ``On multi Hurwitz-zeta function values at rational arguments, Acta Arith. {\bf 107} (2003), 45-67'', 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 zeta-values.

Volume: Volume 24
Published on: January 1, 2001
Submitted on: March 3, 2015
Keywords: functional equation, modular relation, Ramanujan's type formula, special values at rational arguments,Hurwitz zeta-function,[MATH] Mathematics [math]


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