Integral representations of the Riemann zeta function of odd argument

TL;DR

This paper derives integral representations for the Riemann zeta function at odd integers using digamma function expressions, providing new formulas involving elementary functions and confirming some known results.

Contribution

It introduces novel integral formulas for z(2p+1) based on Mikolas's digamma function expression, expanding the analytical tools for studying the zeta function at odd arguments.

Findings

Derived integral representations for z(2p+1)

Provided explicit examples for initial odd arguments

Confirmed some formulas previously obtained by other methods

Abstract

In this article we obtain, using an expression of the digamma function $\psi(x)$ due to Mikolas, integral representations of the zeta function of odd arguments $\zeta(2p+1)$ for any positive value of $p$. The integrand consists of the product of a polynomial by one or two elementary trigonometric functions. Examples for the first values of the argument are given. Some of them were already derived by other methods.