Gram’s Law Fails a Positive Proportion of the Time

It is straightforward to show that all the non-trivial zeroes of the Riemann zeta-function ζ(s) are confined to the critical strip: 0 ≤ R(s) ≤ 1. It is another matter to seek out their precise location. Extending the work of Riemann, Gram introduced a procedure for detecting these zeroes: Gram’s Law. It is known that Gram’s Law fails infinitely often, and that a weaker formulation of Gram’s Law is true infinitely often. This paper extends these results by showing that there is a positive proportion of both failures and (weak) successes. 1 Gram’s Law Connected to ζ(s) are two functions known as the Riemann-Siegel functions, Z(t) = eζ (