Integral representations of the Riemann zeta function of odd argument

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.