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

Abstract

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

arXiv comment7 pages