# 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.

## Key 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.

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Source: https://tomesphere.com/paper/0911.4731