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

S Kanemitsu (1) , Y Tanigawa (2) , M Yoshimoto (3)

1 University ok Kinki
2 Graduate School of Mathematics [Nagoya]
3 RIMS - Research Institute of Mathematical Sciences

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.

Mots clés: Hurwitz zeta-function; special values at rational arguments; Ramanujan's type formula; modular relation; functional equation

Domaines: Mathématiques [math]

S Kanemitsu, Y Tanigawa, M Yoshimoto. On the values of the Riemann zeta-function at rational arguments. Hardy-Ramanujan Journal, 2001, Volume 24 - 2001, pp.9 - 17. ⟨10.46298/hrj.2001.144⟩. ⟨hal-01109799⟩