On Siegel results about the zeros of the auxiliary function of Riemann

TL;DR

This paper revisits Siegel's results on the zeros of the auxiliary function of Riemann, providing complete proofs, clarifying the limit behavior near zeros, and exploring connections to the zeros of the Riemann zeta function.

Contribution

It offers a complete proof of Siegel's results, analyzes the limit behavior of the auxiliary function's zeros, and links these zeros to the zeros of the Riemann zeta function.

Findings

Determination of the limit to the left of zeros with positive imaginary part.

Introduction of a term explaining periodic behavior of zeros.

Connection established between zeros of the auxiliary function and zeta zeros.

Abstract

We state and give complete proof of the results of Siegel about the zeros of the auxiliary function of Riemann $\mathop{\mathcal R}(s)$. We point out the importance of the determination of the limit to the left of the zeros of $\mathop{\mathcal R}(s)$ with positive imaginary part, obtaining the term $-\sqrt{T/2\pi}P(\sqrt{T/2\pi})$ that would explain the periodic behaviour observed with the statistical study of the zeros of $\mathop{\mathcal R}(s)$. We precise also the connection of the position on the zeros of $\mathop{\mathcal R}(s)$ with the zeros of $\zeta(s)$ in the critical line.