Joint distributions of error terms for primes in arithmetic progressions modulo 11
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866909829019205632 |
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| author | Benli, Kübra Martin, Greg Péringuey, Paul |
| author_facet | Benli, Kübra Martin, Greg Péringuey, Paul |
| contents | We provide a formula for the logarithmic density of the set of positive real numbers on which two prime counting functions $ψ(x;q,a)$ and $ψ(x;q,b)$ are simultaneously larger than their asymptotic main terms, as well as a method for calculating the numerical values of such densities with rigorously bounded errors. We apply these formulas to the pairwise races in the case $q=11$, determining which pairs of residues $a$ and $b$ are more or less correlated in this way. The outcomes when $q=11$ provide a deeper mathematical illumination of the "mirror image" and "cyclic ordering" phenomena observed by Bays and Hudson. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_05427 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Joint distributions of error terms for primes in arithmetic progressions modulo 11 Benli, Kübra Martin, Greg Péringuey, Paul Number Theory 11N13 (Primary) 11M26, 11Y35 (Secondary) We provide a formula for the logarithmic density of the set of positive real numbers on which two prime counting functions $ψ(x;q,a)$ and $ψ(x;q,b)$ are simultaneously larger than their asymptotic main terms, as well as a method for calculating the numerical values of such densities with rigorously bounded errors. We apply these formulas to the pairwise races in the case $q=11$, determining which pairs of residues $a$ and $b$ are more or less correlated in this way. The outcomes when $q=11$ provide a deeper mathematical illumination of the "mirror image" and "cyclic ordering" phenomena observed by Bays and Hudson. |
| title | Joint distributions of error terms for primes in arithmetic progressions modulo 11 |
| topic | Number Theory 11N13 (Primary) 11M26, 11Y35 (Secondary) |
| url | https://arxiv.org/abs/2510.05427 |