On the zeros of the Riemann zeta function in the critical strip

Abstract

We describe a computation which shows that the Riemann zeta function ζ ( s ) \zeta (s) has exactly 75,000,000 zeros of the form σ + i t \sigma + it it in the region 0 > t > 32 , 585 , 736.4 0 > t > 32,585,736.4 ; all these zeros are simple and lie on the line σ = 1 / 2 \sigma = 1/2 . (A similar result for the first 3,500,000 zeros was established by Rosser, Yohe and Schoenfeld.) Counts of the number of Gram blocks of various types and the number of failures of "Rosser’s rule" are given.