On the Montgomery-Odlyzko method regarding gaps between zeros of the zeta-function

TL;DR

This paper analyzes the limitations of the Montgomery-Odlyzko method under the Riemann Hypothesis, showing it cannot prove the existence of infinitely many zero pairs closer than 0.5042 times the average spacing.

Contribution

It establishes a new lower bound on the minimal gap size that the Montgomery-Odlyzko method can prove to occur infinitely often.

Findings

The known bound is 0.515396 times the average spacing.

The paper proves the method cannot reach 0.5042 times the average spacing.

This sets a fundamental limit on the method's effectiveness.

Abstract

Assuming the Riemann Hypothesis, it is known that there are infinitely many consecutive pairs of zeros of the Riemann zeta-function within 0.515396 times the average spacing. This is obtained using the method of Montgomery and Odlyzko. We prove that this method can never find infinitely many pairs of consecutive zeros within 0.5042 times the average spacing.