A Reformulation of the Riemann Hypothesis

TL;DR

This paper introduces a new reformulation of the Riemann hypothesis using a pole-free function, (k), that shares the same non-trivial zeros as the zeta function, based on an extended polylogarithm formula.

Contribution

It presents an extended formula for the polylogarithm and a novel reformulation of the Riemann hypothesis via a new function (k) with identical zeros to the zeta function.

Findings

Extended formula for polylogarithm _{k}(e^{z})

Analytic continuation of the zeta function's Dirichlet series

Reformulation of the Riemann hypothesis using (k)

Abstract

We present some novelties on the Riemann zeta function. Using an extended formula created for the polylogarithm in a previous paper, $\mathrm{Li}_{k}(e^{z})$, the zeta function's Dirichlet series is analytically continued from $\Re(k)>1$ to the right half-plane, $\Re(k)>0$, by means of the Dirichlet eta function. More strikingly, we offer a reformulation of the Riemann hypothesis through a zeta's cousin, $\varphi(k)$, a pole-free function defined on the entire complex plane whose non-trivial zeros coincide with those of the zeta function.