Inequities in the Shanks-Renyi Prime Number Race: An asymptotic formula for the densities

TL;DR

This paper derives an asymptotic formula for the densities of prime number races between residue classes, providing insights into their behavior and comparing different classes based on arithmetic properties.

Contribution

It introduces an explicit asymptotic series for the densities elta(q;a,b), expressed via a finite evaluation of a variance related to zeros of Dirichlet L-functions, and offers numerical bounds and comparisons.

Findings

Asymptotic formula for elta(q;a,b) with error smaller than any negative power of q.

Explicit evaluation of variance V(q;a,b) as a finite expression.

Predictions and numerical bounds for densities based on residue class properties.

Abstract

Chebyshev was the first to observe a bias in the distribution of primes in residue classes. The general phenomenon is that if $a$ is a nonsquare\mod q and $b$ is a square\mod q, then there tend to be more primes congruent to $a\mod q$ than $b\mod q$ in initial intervals of the positive integers; more succinctly, there is a tendency for $\pi(x;q,a)$ to exceed $\pi(x;q,b)$. Rubinstein and Sarnak defined $\delta(q;a,b)$ to be the logarithmic density of the set of positive real numbers $x$ for which this inequality holds; intuitively, $\delta(q;a,b)$ is the "probability" that $\pi(x;q,a) > \pi(x;q,b)$ when $x$ is "chosen randomly". In this paper, we establish an asymptotic series for $\delta(q;a,b)$ that can be instantiated with an error term smaller than any negative power of $q$. This asymptotic formula is written in terms of a variance $V(q;a,b)$ that is originally defined as an infinite sum over all nontrivial zeros of Dirichlet $L$-functions corresponding to characters\mod q; we show how $V(q;a,b)$ can be evaluated exactly as a finite expression. In addition to providing the exact rate at which $\delta(q;a,b)$ converges to $\frac12$ as $q$ grows, these evaluations allow us to compare the various density values $\delta(q;a,b)$ as $a$ and $b$ vary modulo $q$; by analyzing the resulting formulas, we can explain and predict which of these densities will be larger or smaller, based on arithmetic properties of the residue classes $a$ and $b\mod q$. For example, we show that if $a$ is a prime power and $a'$ is not, then $\delta(q;a,1) < \delta(q;a',1)$ for all but finitely many moduli $q$ for which both $a$ and $a'$ are nonsquares. Finally, we establish rigorous numerical bounds for these densities $\delta(q;a,b)$ and report on extensive calculations of them.