Zeta Regularized Trigonometric Products Over Zeros Of The Riemann Zeta Function

TL;DR

This paper introduces a new zeta regularized product formula over the zeros of the Riemann zeta function, explores its properties, and discusses its potential implications for the Riemann hypothesis.

Contribution

It presents a novel regularization formula for trigonometric products over zeta zeros and connects it to existing formulas and conjectures.

Findings

Derived a new zeta regularized product formula

Calculated discrepancies of the regularized products

Linked the formula to the Kimoto-Wakayama formula and Riemann hypothesis

Abstract

We prove a novel zeta regularized product formula concerning regularization of trigonometric products over non-trivial zeros of the Riemann zeta function. Furthermore, we calculate the discrepancies of such regularized products. In special cases, our formula reduces to the Kimoto-Wakayama formula. A conjectural relationship between such products and a weak Riemann hypothesis is speculated.