The Alternative Hypothesis for Zeros of the Riemann Zeta-Function
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866913991904722944 |
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| author | Baluyot, Siegfred Alan C. Goldston, Daniel Alan Suriajaya, Ade Irma Turnage-Butterbaugh, Caroline L. |
| author_facet | Baluyot, Siegfred Alan C. Goldston, Daniel Alan Suriajaya, Ade Irma Turnage-Butterbaugh, Caroline L. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_10857 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Alternative Hypothesis for Zeros of the Riemann Zeta-Function Baluyot, Siegfred Alan C. Goldston, Daniel Alan Suriajaya, Ade Irma Turnage-Butterbaugh, Caroline L. Number Theory 11M06, 11M26 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. |
| title | The Alternative Hypothesis for Zeros of the Riemann Zeta-Function |
| topic | Number Theory 11M06, 11M26 |
| url | https://arxiv.org/abs/2508.10857 |