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

TL;DR

This paper establishes new relationships between values of the Hurwitz zeta function at integers and rational points and generalized harmonic numbers using Euler transformations, providing explicit formulas and generating functions.

Contribution

It introduces novel series representations linking Hurwitz zeta values to generalized harmonic numbers and derives their generating functions.

Findings

Proves a formula for ζ(k,1) involving generalized harmonic numbers.

Derives the generating function for the alternating zeta sum.

Provides rapidly converging series for zeta function values.

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}. $