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 be a real number with 12< 1, and let N (, T) be the number of zeros =+i of R (s) with 1 and 0< T. Then we prove (, T) T^32- (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 ^-s/2 (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 (s) in the critical line.