A family of integrals related to values of the Riemann zeta function

TL;DR

This paper introduces new integral representations for both even and odd positive integer values of the Riemann zeta function, linking these values to specific integrals and harmonic sums.

Contribution

It provides novel integral formulas for zeta(2p) and zeta(2p+1), enhancing understanding of their structure and relationships.

Findings

Derived integral representations for ζ(2p) and ζ(2p+1)

Expressed ζ(2p+1) using integrals and harmonic sums

Simplified the representation of ζ(2p+1) through integral forms

Abstract

We propose a relation between values of the Riemann zeta function $\zeta$ and a family of integrals. This results in an integral representation for $\zeta(2p)$, where $p$ is a positive integer, and an expression of $\zeta(2p+1)$ involving one of the above mentioned integrals together with a harmonic-number sum. Simplification of the latter eventually leads to an integral representation of $\zeta(2p + 1)$.