Logarithmic Fourier integrals for the Riemann Zeta Function

TL;DR

This paper develops new integral formulas involving the Riemann zeta function using symmetric Poisson-Schwarz formulas, providing explicit representations that relate to prime number distribution and zeroes on the critical line.

Contribution

It introduces novel factorization theorems for the zeta function and variants of the Balazard-Saias-Yor theorem using integral formulas along the critical line.

Findings

Derived explicit formulas involving zeta zeroes and prime distribution

Established new factorization theorems for the Riemann zeta function

Provided integral representations using Blaschke products

Abstract

We use symmetric Poisson-Schwarz formulas for analytic functions $f$ in the half-plane ${Re}(s)>\frac12$ with $\bar{f(\bar{s})}=f(s)$ in order to derive factorisation theorems for the Riemann zeta function. We prove a variant of the Balazard-Saias-Yor theorem and obtain explicit formulas for functions which are important for the distribution of prime numbers. In contrast to Riemann's classical explicit formula, these representations use integrals along the critical line ${Re}(s)=\frac12$ and Blaschke zeta zeroes.