On the number of zeros of R (s)

We prove that the number of zeros =+i of R (s) with 0< T is given by (T) =T42-T4-12T{2}+O (T^2/5² 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.