The zeros of the Riemann zeta-function

Abstract

1―It is well known that the distribution of the zeroes of the Riemann zeta-function ζ( s ) = ∞ Σ n=1 1/ n 8 ( s = σ + it ) plays a fundamental part in the theory of prime numbers. It was conjectured by Riemann that all the complex zeroes of ζ( s ) lie on the line σ = 1/2, but this hypothesis has never been proved or disproved. It is therefore natural to enquiry how far the hypothesis is supported by numerical calculations. The most extensive calculations of this kind have been undertaken by Gram, Backlund, and Hutchinson. The final result obtained by Hutchinson is that ζ( s ) has 138 zeroes on σ = 1/2 between t = 0 and t = 300, and no other zeroes between these values of t .