The Shanks-Rényi prime number race with many contestants

Under certain plausible assumptions, M. Rubinstein and P. Sarnak solved the Shanks--Rényi race problem, by showing that the set of real numbers $x\geq 2$ such that $π(x;q,a_1)>π(x;q,a_2)>...>π(x;q,a_r)$ has a positive logarithmic density $δ_{q;a_1,...,a_r}$. Furthermore, they established that if $r$ is fixed, $δ_{q;a_1,...,a_r}\to 1/r!$ 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 $δ_{q;a_1,...,a_r}=o(1)$ in this case. Among our results, we prove that $δ_{q;a_1,...,a_r}\sim 1/r!$ 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 $\log q\leq r\leq ϕ(q)$, $δ_{q;a_1,...,a_r}\ll_ε q^{-1+ε}$.

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