On connection between values of Riemann zeta function at integers and generalized harmonic numbers

Abstract

Using Euler transformation of series we relate values of Hurwitz zeta function at integer and rational values of arguments to certain rapidly converging series where some generalized harmonic numbers appear. The form of these generalized harmonic numbers carries information about the values of the arguments of Hurwitz function. In particular we prove: $\forall k\in \mathbb{N}:$ $\zeta (k,1)\allowbreak =\allowbreak \frac{2^{k-1}}{2^{k-1}-1}% \sum_{n=1}^{\infty }\frac{H_{n}^{(k-1)}}{n2^{n}},$ where $H_{n}^{(k)}$ are defined below generalized harmonic numbers. Further we find generating function of the numbers $\hat{\zeta}(k)=\sum_{j=1}^{\infty }(-1)^{j-1}/j^{k}. $