Infinite Product of the Riemann auxiliary function

TL;DR

This paper derives an infinite product representation for the Riemann auxiliary function and investigates how its zeros influence the phase function, shedding light on the zeros of the Riemann zeta function on the critical line.

Contribution

It introduces a new infinite product formula for the auxiliary function and explores its zeros' impact on the phase and the zeros of the zeta function.

Findings

The phase function $\omega(t)$ is linked to the zeros of $\mathcal R(s)$ and $\zeta(s)$.

Zeros of $\mathcal R(s)$ influence the behavior of $\omega(t)$.

The relationship between zeros of $\mathcal R(s)$ and $\zeta(s)$ is explicitly characterized.

Abstract

We obtain the product for the auxiliary function $\mathop{\mathcal R}(s)$ and study some related functions as its phase $\omega(t)$ at the critical line. The function $\omega(t)$ determines the zeros of $\zeta(s)$ on the critical line. We study the influence of the zeros of $\mathop{\mathcal R}(s)$ on $\omega(t)$. Thus, the relationship between the zeros of $\mathop{\mathcal R}(s)$ and those of $\zeta(s)$ is determined.