Joint Distributions of Error Terms for Primes in Arithmetic Progressions Modulo 11
Abstract
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.
Fields of Study & Subjects
Analytic Number Theory Research
Cryptography and Residue Arithmetic
History and Theory of Mathematics
Publication details
- Journal / source
- Experimental Mathematics
- Publisher
- Informa UK Limited
- Pages
- 1–19
- Publication date
- 17 September 2026
- Language
- EN