An integral representation for $\zeta(4)$

TL;DR

This paper introduces a new integral representation for the Riemann zeta function at 4, derived via polylogarithm relations, and discusses potential generalizations to other even arguments.

Contribution

It presents a novel integral formula for (4) and explores its extension to all even positive integers of the zeta function.

Findings

Derived an explicit integral representation for (4)

Connected the representation to polylogarithm relations

Suggested a framework for generalizing to other even arguments

Abstract

In this note, we propose an integral representation for $\zeta(4)$, where $\zeta$ is the Riemann zeta function. The corresponding expression is obtained using relations for polylogarithms. A possible generalization to any even argument of the zeta function is considered.