A Wasserstein metric approach to generalized Skewes' numbers. I. Prime number races
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866910161480712192 |
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| author | Bailleul, Alexandre Hayani, Mounir Untrau, Théo |
| author_facet | Bailleul, Alexandre Hayani, Mounir Untrau, Théo |
| contents | We study generalized Skewes' numbers, which are the locations of the first sign change between two comparable prime counting functions. In the context of the race between quadratic residues and quadratic nonresidues, we construct sequences of highly composite moduli $q$ such that those Skewes' numbers grow very rapidly in some sense. This disproves unconditionally a conjecture of Fiorilli. In the other direction, assuming the Generalized Riemann Hypothesis and an effective linear independence hypothesis, we establish conditional upper bounds for generalized Skewes' numbers. Our approach relies on a quantitative Kronecker-Weyl theorem formulated in terms of the $1$-Wasserstein metric to obtain explicit rates for the convergence to the limiting distributions in these races. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_20093 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Wasserstein metric approach to generalized Skewes' numbers. I. Prime number races Bailleul, Alexandre Hayani, Mounir Untrau, Théo Number Theory 11N13, 11M26 (Primary), 60B10, 60F10, 11K70 (Secondary) We study generalized Skewes' numbers, which are the locations of the first sign change between two comparable prime counting functions. In the context of the race between quadratic residues and quadratic nonresidues, we construct sequences of highly composite moduli $q$ such that those Skewes' numbers grow very rapidly in some sense. This disproves unconditionally a conjecture of Fiorilli. In the other direction, assuming the Generalized Riemann Hypothesis and an effective linear independence hypothesis, we establish conditional upper bounds for generalized Skewes' numbers. Our approach relies on a quantitative Kronecker-Weyl theorem formulated in terms of the $1$-Wasserstein metric to obtain explicit rates for the convergence to the limiting distributions in these races. |
| title | A Wasserstein metric approach to generalized Skewes' numbers. I. Prime number races |
| topic | Number Theory 11N13, 11M26 (Primary), 60B10, 60F10, 11K70 (Secondary) |
| url | https://arxiv.org/abs/2603.20093 |