Density theorems for Riemann's auxiliary function

We prove a density theorem for the auxiliar function R(s) found by Siegel in Riemann papers. Let α be a real number with 12< α≤ 1, and let N(α,T) be the number of zeros ρ=β+iγ of R(s) with 1≥ β≥α and 0<γ≤ T. Then we prove \[N(α,T) T32-α(log T)^3.\] Therefore, most of the zeros of R(s) are near the critical line or to the left of that line. The imaginary line for π-s/2Γ(s/2) R(s) passing through a zero of R(s) near the critical line frequently will cut the critical line, producing two zeros of ζ(s) in the critical line.