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arXiv · 2510.05427

Joint distributions of error terms for primes in arithmetic progressions modulo 11

Kübra Benli·Greg Martin·Paul Péringuey
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Abstract

We provide a formula for the logarithmic density of the set of positive real numbers on which two prime counting functions $\psi(x;q,a)$ and $\psi(x;q,b)$ 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.

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Kübra Benli, Greg Martin, Paul Péringuey. 2025-10-06. Joint distributions of error terms for primes in arithmetic progressions modulo 11. https://arxiv.org/abs/2510.05427

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