Integral representations for $\zeta(3)$ with the inverse sine function

TL;DR

This paper introduces four novel integral representations of the Riemann zeta function at 3 using inverse sine and Wallis integrals, also deriving a local integral form for the trilogarithm.

Contribution

It presents new integral formulas for ζ(3) based on inverse sine and Wallis integrals, extending previous representations and including a local form for the trilogarithm.

Findings

Four new integral representations of ζ(3)

A local integral representation for the trilogarithm

Extension of previous integral formulas

Abstract

We show four new integral representations for $\zeta(3)$ as a reformulation of Ewell (1990) and Yue-Williams (1993) with the inverse sine function and Wallis integral. As a consequence, we also show a local integral representation for the trilogarithm function.