The Riemann Hypothesis: A Qualitative Characterization of the Nontrivial Zeros of the Riemann Zeta Function Using Polylogarithms

TL;DR

This paper introduces a new parametrized series representation of the Riemann zeta function involving polylogarithms, providing a novel qualitative characterization of its nontrivial zeros through convergence properties and functional equations.

Contribution

It formulates a new uniformly absolutely convergent series Z(s, x) for ζ(s) using polylogarithms, offering a fresh perspective on the zeros of ζ(s).

Findings

New series representation of ζ(s) involving polylogarithms.

Expresses ζ(s) in terms of an integral of Li_s(z).

Provides a qualitative characterization of nontrivial zeros.

Abstract

We formulate a parametrized uniformly absolutely globally convergent series of $\zeta$(s) denoted by Z(s, x). When expressed in closed form, it is given by Z(s, x) = (s -- 1)$\zeta$(s) + 1 x Li s z z -- 1 dz, where Li s (x) is the polylogarithm function. As an immediate first application of the new parametrized series, a new expression of $\zeta$(s) follows: (s -- 1)$\zeta$(s) = -- 1 0 Li s z z -- 1 dz. As a second important application, using the functional equation and exploiting uniform convergence of the series defining Z(s, x), we have for any non-trivial zero s