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

2025/10/06 by Kübra Benli̇, Greg Martin, Benli, Kübra +3
Computer Science · Mathematics · #11N13 (Primary) 11M26 #11Y35 (Secondary) #Analytic Number Theory Research #Cryptography and Residue Arithmetic #FOS: Mathematics #History and Theory of Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2510.05427

openalex publication_date 2025/10/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Inclusive prime number races
• Limiting distributions of the classical error terms of prime number theory
• Inequities in the Shanks-Renyi Prime Number Race: An asymptotic formula\n for the densities
• Biases in the Shanks-Renyi Prime Number Race

Related

• Correlations of error terms for weighted prime counting functions
• The error term in counting prime pairs
• The distribution of consecutive prime biases and sums of sawtooth random\n variables
• The Error Term In The Primes Counting Function
• Inequities in the Shanks-Renyi Prime Number Race: An asymptotic formula\n for the densities