A formula for $\zeta$(2n + 1) and some related expressions

TL;DR

The paper derives integral representations for the Riemann zeta function at odd integers using polylogarithmic identities, providing new formulas and properties for these values and related integrals.

Contribution

It introduces novel integral formulas for ζ(2n+1) and explores properties and closed-form expressions of these integrals, advancing understanding of odd zeta values.

Findings

Integral representations for ζ(2n+1) in n-dimensional hypercubes

Properties of functions involving logarithms when variables are raised to powers

Closed-form expressions for specific related integrals when n=2

Abstract

Using a polylogarithmic identity, we express the values of $\zeta$ at odd integers $2n+1$ as integrals over unit $n-$dimensional hypercubes of simple functions involving products of logarithms. We also prove a useful property of those functions as some of their variables are raised to a power. In the case $n=2$, we prove two closed-form expressions concerning related integrals. Finally, another family of related one-dimensional integrals is studied.