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

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 $a\mathrm{mod}q$ than $b\mathrm{mod}q$ 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 $\frac{1}{2}$ 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 $b\mathrm{mod}q$. For example, we show that if $a$ is a prime power and $a^{\prime }$ is not, then $\delta (q;a,1)<\delta (q;a^{\prime },1)$ for all but finitely many moduli $q$ for which both $a$ and $a^{\prime }$ are nonsquares. Finally, we establish rigorous numerical bounds for these densities $\delta (q;a,b)$ and report on extensive calculations of them.

arXiv comment76 pages. Corrected a numerical error in Proposition 2.14 and its consequences in Section 5. Paper has been accepted to Crelle