Limiting properties of the distribution of primes in an arbitrarily large number of residue classes

TL;DR

This paper extends the understanding of prime distribution in multiple residue classes, showing that certain inequalities among prime counts have a well-defined density under specific hypotheses.

Contribution

It generalizes prime number race results to arbitrarily many residue classes, using methods applicable to classical and function field cases.

Findings

The set of x satisfying the inequalities has a logarithmic density.

Results depend on the generalized Riemann Hypothesis and linear independence hypothesis.

Applicable to both classical and function field analogues.

Abstract

We generalize current known distribution results on Shanks--R\'enyi prime number races to the case where arbitrarily many residue classes are involved. Our method handles both the classical case that goes back to Chebyshev and function field analogues developed in the recent years. More precisely, let $\pi(x;q,a)$ be the number of primes up to $x$ that are congruent to $a$ modulo $q$. For a fixed integer $q$ and distinct invertible congruence classes $a_0,a_1,\ldots,a_D$, assuming the generalized Riemann Hypothesis and a weak version of the linear independence hypothesis, we show that the set of $x$ real for which the inequalities $\pi(x;q,a_0)>\pi(x;q,a_1)> \ldots >\pi(x;q,a_D)$ are simultaneously satisfied admits a logarithmic density.