The Alternative Hypothesis for Zeros of the Riemann Zeta-Function

2025/08/14 by Baluyot, Siegfred Alan C., Goldston, Daniel Alan, Suriajaya, Ade Irma +1
#11M06 #11M26 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2508.10857

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.

Citations

• Pair Correlation of Zeros of the Riemann Zeta Function I: Proportions of Simple Zeros and Critical Zeros
• Pair Correlation Estimates for the Zeros of the Zeta Function via\n Semidefinite Programming

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