The Riemann zeta in terms of the dilogarithm

TL;DR

This paper presents new integral representations of the Riemann zeta function using the dilogarithm and Clausen functions, linking classical zeta analysis with special functions and Mellin transforms.

Contribution

It introduces novel Mellin transform representations of the zeta function involving the dilogarithm and Clausen functions, along with generalized M"untz-type formulas.

Findings

Derived integral representations of ζ(s) in terms of dilogarithm and Clausen functions.

Established connections between zeta series and gamma functions.

Provided explicit Mellin transform computations for special test functions.

Abstract

We give a representation of the classical Riemann $\zeta$-function in the half plane $\Re s>0$ in terms of a Mellin transform involving the real part of the dilogarithm function with an argument on the unit circle (associated Clausen $Gl_2$-function). We also derive corresponding representations involving the derivatives of the $Gl_2$-function. A generalized symmetrized M\"untz-type formula is also derived. For a special choice of test functions it connects to our integral representation of the $\zeta$-function, providing also a computation of a concrete Mellin transform. Certain formulae involving series of zeta functions and gamma functions are also derived.