Integral representations of the Legendre chi function

TL;DR

This paper derives integral representations of the Legendre chi function using elementary methods, which also lead to new insights into the Riemann zeta and Dirichlet beta functions for specific arguments.

Contribution

It introduces new integral representations of the Legendre chi function and connects these to known results for the Riemann zeta and Dirichlet beta functions.

Findings

Derived integral representations valid for |z|<1 and Re(s)>1

Connected representations to values of zeta and beta functions

Provided elementary proofs for these integral formulas

Abstract

We, by making use of elementary arguments, deduce integral representations of the Legendre chi function $\chi_{s}(x)$ valid for $|z|<1$ and $\Re(s)>1$. Our earlier established results on the integral representations for the Riemann zeta function $\zeta(2 n+1)$ and the Dirichlet beta function $\beta(2 n)$ ,$ n\in\mathbb{N}$, are direct consequence of these representations.