# The Alternative Hypothesis for Zeros of the Riemann Zeta-Function

Source: [The Alternative Hypothesis for Zeros of the Riemann Zeta-Function](<https://www.sota2.com/research/paper/68fa5e051e209131ce9dea7f>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-25T18:32:57.339Z

Authors: Siegfred Alan C. Baluyot, Daniel Alan Goldston, Ade Irma Suriajaya, Caroline L. Turnage-Butterbaugh

Published: 2025-08-14T17:29:05.000Z

## 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.

## Recorded benchmark results

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/2508.10857>)

[PDF](<https://arxiv.org/pdf/2508.10857.pdf>)