Gram's Law Fails a Positive Proportion of the Time

TL;DR

This paper demonstrates that Gram's Law, which relates to the zeros of the zeta-function, fails in a positive proportion of intervals, and a weaker version holds in a positive proportion as well.

Contribution

It extends Titchmarsh's work by proving that Gram's Law fails in a positive proportion of intervals and introduces a weaker form that also holds in a positive proportion.

Findings

A positive proportion of Gram intervals violate Gram's Law.

A weaker notion of Gram's Law is valid over a positive proportion of intervals.

The results provide new insights into the distribution of zeros of the zeta-function.

Abstract

This paper extends the work done by Titchmarsh on Gram's Law (an attempt to locate the zeroes of the zeta-function on the critical line). Herewith it is shown that a positive proportion of Gram intervals violate Gram's Law; and also that a weaker notion of Gram's Law is valid over a positive proportion of intervals.