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 $x\ge 2$ such that $\pi (x;q,a_1)>\pi (x;q,a_2)>\mathrm{.}\mathrm{.}\mathrm{.}>\pi (x;q,a_r)$ has a positive logarithmic density ${\delta }_{q;a_1,\mathrm{.}\mathrm{.}\mathrm{.},a_r}$. Furthermore, they established that if $r$ is fixed, ${\delta }_{q;a_1,\mathrm{.}\mathrm{.}\mathrm{.},a_r}\to 1\mathrm{/}r!$ as $q\to \mathrm{\infty }$. 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 ${\delta }_{q;a_1,\mathrm{.}\mathrm{.}\mathrm{.},a_r}=o(1)$ in this case. Among our results, we prove that ${\delta }_{q;a_1,\mathrm{.}\mathrm{.}\mathrm{.},a_r}\sim 1\mathrm{/}r!$ in the region $r=o(\sqrt{\log q})$ as $q\to \mathrm{\infty }$. We also bound the order of magnitude of these densities beyond this range of $r$. For example, we show that when $\log q\le r\le \varphi (q)$, ${\delta }_{q;a_1,\mathrm{.}\mathrm{.}\mathrm{.},a_r}{\ll }_{ϵ}{q}^{-1+ϵ}$.