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arXiv · 2508.10857

The Alternative Hypothesis for Zeros of the Riemann Zeta-Function

Siegfred Alan C. Baluyot·Daniel Alan Goldston·Ade Irma Suriajaya·Caroline L. Turnage-Butterbaugh
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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.

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Siegfred Alan C. Baluyot, Daniel Alan Goldston, Ade Irma Suriajaya, Caroline L. Turnage-Butterbaugh. 2025-08-14. The Alternative Hypothesis for Zeros of the Riemann Zeta-Function. https://arxiv.org/abs/2508.10857

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