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

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 $\log q\leq r\leq \phi(q)$, $\delta_{q;a_1,...,a_r}\ll_{\epsilon} q^{-1+\epsilon}$.