Joint distributions of error terms for primes in arithmetic progressions modulo 11
Kübra Benli, Greg Martin, Paul Péringuey
TL;DR
This work formulates a rigorous, Fourier-analytic method to quantify the joint positivity of normalized prime-counting error terms in arithmetic progressions modulo $q$, focusing on $q=11$. By modeling the limiting distribution of the vector of error terms with random Dirichlet-L-function zeros, it derives an explicit double-integral formula for the first-quadrant density $\delta^{++}_a$ and provides a framework for computing these densities with controlled errors. The authors carry out detailed numerical computations for $q=11$, obtaining precise values for $\delta_a^{++}$ and confirming mirror-image and cyclic-ordering phenomena in the corresponding prime races. An explanatory model highlights the role of a single low-lying zero of a Dirichlet $L$-function as the primary mediator of the cyclic behavior, linking analytic structure to observed statistical patterns. The combination of rigorous numerics and a probabilistic model advances understanding of correlations among prime-counting functions in residue classes and sharpens the predictive picture of prime races 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.