A further expansion of the Riemann \(\zeta\)-function
The author continues the investigations on the expansion of the Hurwitz zeta-function obtained by \textit{S. Kanemitsu, M. Katsurada} and \textit{M. Yoshimoto} [Aequationes Math. 59, No. 1--2, 1--19 (2000; Zbl 1059.11050)], \textit{S. Kanemitsu, H. Kumagai, H. M. Srivastava} and \textit{M.~Yoshimoto} [Appl. Math. Comput. 154, No. 3, 641--664 (2004; Zbl 1130.11049)], and \textit{K. Ueno} and \textit{M. Nishizawa} [Quantum groups: formalism and applications. Proceedings of the XXX Karpacz winter school of theoretical physics, Karpacz, Poland, February 14--26, 1994. Warszawa: PWN-Polish Scientific Publishers, 115--126 (1995; Zbl 0874.17006)] and proves a similar result for the Riemann zeta-function. More precisely, the author gives the following explicit expansion \[ \zeta(s)={1\over s-1}+{1\over 2}+(-1)^s{s\over 2\pi i}{1\over \Gamma(s)}\sum_{\nu\neq0}{1\over \nu}\left(\sum_{n=0}^\infty{\psi(1-s+n)-\log(-2\pi i \nu)\over \Gamma (1-s+n)}(-2\pi i\nu)^n\right) \] with the diagamma function \[ \psi(1-s+n)=-\gamma-\sum_{n=2}^\infty\zeta(n)(s-n)^{n-1}, | s-n| <1, \] where \(\gamma\) is the Euler-Mascheroni constant.
scientific article; zbMATH DE number 5163963