The Shanks-R\'enyi prime number race with many contestants

TL;DR

This paper studies the distribution of prime number races with many contestants, showing that the probability of a specific ordering tends to zero as the number of contestants grows, and establishing asymptotic behaviors in different regimes.

Contribution

It extends the understanding of prime number races to large numbers of contestants, proving asymptotic formulas and bounds for the associated densities as the number of contestants increases.

Findings

For r = o(√log q), δ_{q;a_1,...,a_r} ~ 1/r!

When log q ≤ r ≤ φ(q), δ_{q;a_1,...,a_r} ≪ q^{-1+ε}

The densities tend to zero as the number of contestants grows large.

Abstract

Under certain plausible assumptions, M. Rubinstein and P. Sarnak solved the Shanks--R\'enyi race problem, by showing that the set of real numbers $x\geq 2$ such that $\pi(x;q,a_1)>\pi(x;q,a_2)>...>\pi(x;q,a_r)$ has a positive logarithmic density $\delta_{q;a_1,...,a_r}$. Furthermore, they established that if $r$ is fixed, $\delta_{q;a_1,...,a_r}\to 1/r!$ as $q\to \infty$. In this paper, we investigate the size of these densities when the number of contestants $r$ tends to infinity with $q$. In particular, we deduce a strong form of a recent conjecture of A. Feuerverger and G. Martin which states that $\delta_{q;a_1,...,a_r}=o(1)$ in this case. Among our results, we prove that $\delta_{q;a_1,...,a_r}\sim 1/r!$ in the region $r=o(\sqrt{\log q})$ as $q\to\infty$. We also bound the order of magnitude of these densities beyond this range of $r$. For example, we show that when $\log q\leq r\leq \phi(q)$, $\delta_{q;a_1,...,a_r}\ll_{\epsilon} q^{-1+\epsilon}$.