On the number of zeros of R(s)

We prove that the number of zeros =β+iγ of R(s) with 0<γ≤ T is given by \[N(T)=T/4π/2π-T/4π-12√T/2π+O(T2/5log^2 T).\] Here R(s) is the function that Siegel found in Riemann's papers. Siegel related the zeros of R(s) to the zeros of Riemann's zeta function. Our result on $N(T)$ improves the result of Siegel.