Density theorems for Riemann's auxiliary function

Abstract

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

arXiv comment14 pages, 4 figures