Polylogarithm approaches to Riemann Zeta function zeroes

TL;DR

This paper explores polylogarithm identities near Riemann zeta zeroes, deriving new formulas and infinite product identities that relate to trivial and critical line zeroes.

Contribution

It introduces novel formulas involving polylogarithms and Euler-Zagier sums that connect to the zeroes of the Riemann zeta function.

Findings

New formulas for trivial zeroes and critical line zeroes

Infinite product identities derived from Euler-Zagier sums

Enhanced understanding of zeta function zeroes near critical points

Abstract

We use visible point vector identities to examine polylogarithms in the neighbourhood of the Riemann zeta function zeroes. New formulas limiting to the trivial zeroes and to the critical line on the zeta function are given. Similar results arise from Euler-Zagier sums given in Bailey, Borwein and Crandall [4] providing new infinite product identities.