On the number of zeros of $\mathop{\mathcal R}(s)$

TL;DR

This paper establishes an improved asymptotic formula for counting the zeros of Siegel's function al R(s), which is related to the zeros of the Riemann zeta function, refining Siegel's earlier results.

Contribution

The paper provides a sharper asymptotic estimate for the number of zeros of al R(s), advancing understanding of its zero distribution and its connection to the Riemann zeta function.

Findings

Derived an explicit asymptotic formula for zeros count N(T).

Improved upon Siegel's previous estimates for zeros of al R(s).

Enhanced understanding of the zero distribution related to the Riemann zeta function.

Abstract

We prove that the number of zeros $\varrho=\beta+i\gamma$ of $\mathop{\mathcal R}(s)$ with $0<\gamma\le T$ is given by \[N(T)=\frac{T}{4\pi}\log\frac{T}{2\pi}-\frac{T}{4\pi}-\frac12\sqrt{\frac{T}{2\pi}}+O(T^{2/5}\log^2 T).\] Here $\mathop{\mathcal R}(s)$ is the function that Siegel found in Riemann's papers. Siegel related the zeros of $\mathop{\mathcal R}(s)$ to the zeros of Riemann's zeta function. Our result on $N(T)$ improves the result of Siegel.