The Alternative Hypothesis for Zeros of the Riemann Zeta-Function

TL;DR

This paper explores a formulation of the Alternative Hypothesis regarding the spacing of zeros of the Riemann zeta-function, deriving constraints on zero densities and implications for zero simplicity under the Riemann Hypothesis.

Contribution

It introduces new density constraints for zeros spaced at specific multiples of the average, linking the Alternative Hypothesis to zero simplicity.

Findings

Constraints on the density of zero pairs at specific spacings

Restrictions on the density of multiple zeros

Implication that a stronger hypothesis implies the Essential Simplicity Hypothesis

Abstract

In 2016, the first-named author introduced a formulation of the Alternative Hypothesis that assumes that consecutive zeros of the Riemann zeta-function are spaced at multiples of half of the average spacing, but does not assume that the zeros are simple. In this paper, we assume the Riemann Hypothesis and a similar formulation of the Alternative Hypothesis, and for each integer $k$ we obtain constraints on the density of pairs of zeros whose normalized differences are at $k/2$ times the average spacing. These constraints, in turn, restrict the density of (possible) multiple zeros. We also formulate a stronger version of the Alternative Hypothesis and show that it implies the Essential Simplicity Hypothesis.