On the number of zeros of $<mi\; mathvariant="script">R</mi>(s)$

We prove that the number of zeros $\varrho =\beta +i\gamma$ of $(s)$ with $0<\gamma \le T$ is given by $$N(T)=\frac{T}{4\pi }\log \frac{T}{2\pi }-\frac{T}{4\pi }-\frac{1}{2}\sqrt{\frac{T}{2\pi }}+O(T^{2\mathrm{/}5}{\log }^{2}T)\mathrm{.}$$ Here $(s)$ is the function that Siegel found in Riemann’s papers. Siegel related the zeros of $(s)$ to the zeros of Riemann’s zeta function. Our result on $N(T)$ improves the result of Siegel.