Density theorems for Riemann's auxiliary function

TL;DR

This paper establishes a density theorem for Siegel's auxiliary function, showing most of its zeros lie near or to the left of the critical line, with implications for the zeros of the Riemann zeta function.

Contribution

It provides a new density estimate for the zeros of Siegel's auxiliary function, enhancing understanding of their distribution relative to the critical line.

Findings

Most zeros of the auxiliary function are near or to the left of the critical line.

The number of zeros with real part ≥ α grows slower than T^{3/2 - α} (log T)^3.

Zeros near the critical line influence the distribution of zeros of the Riemann zeta function.

Abstract

We prove a density theorem for the auxiliar function $\mathop{\mathcal R}(s)$ found by Siegel in Riemann papers. Let $\alpha$ be a real number with $\frac12< \alpha\le 1$, and let $N(\alpha,T)$ be the number of zeros $\rho=\beta+i\gamma$ of $\mathop{\mathcal R}(s)$ with $1\ge \beta\ge\alpha$ and $0<\gamma\le T$. Then we prove \[N(\alpha,T)\ll T^{\frac32-\alpha}(\log T)^3.\] Therefore, most of the zeros of $\mathop{\mathcal R}(s)$ are near the critical line or to the left of that line. The imaginary line for $\pi^{-s/2}\Gamma(s/2)\mathop{\mathcal R}(s)$ passing through a zero of $\mathop{\mathcal R}(s)$ near the critical line frequently will cut the critical line, producing two zeros of $\zeta(s)$ in the critical line.