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

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.