A New Representation of the Riemann Zeta Function

TL;DR

This paper introduces a novel representation of the Riemann zeta function as a limit of structured double sums, offering new theoretical insights and identities involving maximum functions and harmonic series.

Contribution

It presents a new representation of the zeta function using double sums and identities involving maximum functions, connecting classical analysis with modern summation methods.

Findings

New representation of the zeta function as a limiting difference of double sums

Derived identities involving maximum functions and additive terms

Links between harmonic series, polygamma functions, and the zeta function

Abstract

In this paper,we develop a novel representation of the zeta function expressed as the limiting difference between two structured double sums. This approach leads to a new and elegant identity involving maximum functions and additive terms, providing theoretical insights. The derivation relies on generalized harmonic series and polygamma functions, linking classical analysis with contemporary summation techniques.