Logarithmic densities of prime number races between the cosets of a subgroup of index 4 modulo 24

For each subgroup H of index 4 of (Z/24Z)^× and each nontrivial coset S of H, this note gives the logarithmic density δ(24;S,H) of the set of real numbers x for which there are more primes up to x in the residue classes of S than in those of H (21 races), together with 14 further races modulo 24 in an appendix. The values are those of the Rubinstein–Sarnak model, so they are conditional on the Generalized Riemann Hypothesis and on the Linear Independence Hypothesis; they are numerical values, not rigorous bounds. The note explains why I believe each value is correctly rounded to five decimals, compares fifteen of them with previously published values, and includes a short PARI/GP script (verify_densities_mod24.gp, with its output verify_densities_mod24_T1000.out) that recomputes every value from scratch with public tools. SHA256SUMS.txt lists the SHA-256 checksums of the three files. Use of AI tools: I used AI assistance, under my supervision, in drafting this note and in carrying out and checking all the mathematical computations.

Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-11-06
DOI
https://doi.org/10.5281/zenodo.24020024
Primary Topic
Analytic Number Theory Research
Type
preprint
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preprint

Logarithmic densities of prime number races between the cosets of a subgroup of index 4 modulo 24

Leandro Hector Gallego
1 links
Zenodo (CERN European Organization for Nuclear Research)
Analytic Number Theory Research
preprint

Logarithmic densities of prime number races between the cosets of a subgroup of index 4 modulo 24

Leandro Hector Gallego
preprint en

Abstract

For each subgroup H of index 4 of (Z/24Z)^× and each nontrivial coset S of H, this note gives the logarithmic density δ(24;S,H) of the set of real numbers x for which there are more primes up to x in the residue classes of S than in those of H (21 races), together with 14 further races modulo 24 in an appendix. The values are those of the Rubinstein–Sarnak model, so they are conditional on the Generalized Riemann Hypothesis and on the Linear Independence Hypothesis; they are numerical values, not rigorous bounds. The note explains why I believe each value is correctly rounded to five decimals, compares fifteen of them with previously published values, and includes a short PARI/GP script (verify_densities_mod24.gp, with its output verify_densities_mod24_T1000.out) that recomputes every value from scratch with public tools. SHA256SUMS.txt lists the SHA-256 checksums of the three files. Use of AI tools: I used AI assistance, under my supervision, in drafting this note and in carrying out and checking all the mathematical computations.

Zenodo (CERN European Organization for Nuclear Research)
Analytic Number Theory Research
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