Mathematical Research Letters

Volume 19 (2012)

Number 3

The Shanks–Rényi prime number race with many contestants

Pages: 649 – 666



Youness Lamzouri (Department of Mathematics and Statistics, York University, Toronto, Ontario, Canada)


Under certain plausible assumptions, M. Rubinstein and P.Sarnak solved the Shanks–Rényi race problem by showingthat the set of real numbers $x\geq 2$ such that$\pi(x;q,a_1)>\pi(x;q,a_2)>\cdots>\pi(x;q,a_r)$ has apositive logarithmic density $\delta_{q;a_1,\dots,a_r}$.Furthermore, they established that if $r$ is fixed,$\delta_{q;a_1,\dots,a_r}\to 1/r!$ as $q\to \infty$. Inthis paper, we investigate the size of these densities whenthe number of contestants $r$ tends to infinity with $q$.In particular, we deduce a strong form of a recentconjecture of Feuerverger and Martin which states that$\delta_{q;a_1,\dots,a_r}=o(1)$ in this case. Among ourresults, we prove that $\delta_{q;a_1,\dots,a_r}\sim 1/r!$in the region $r=o(\sqrt{\log q})$ as $q\to\infty$. We alsobound the order of magnitude of these densities beyond thisrange of $r$. For example, we show that when $\log q\leqr\leq \phi(q)$, $\delta_{q;a_1,\dots,a_r}\ll_{\epsilon}q^{-1+\epsilon}$.


Shanks–Rényi race problem, primes in arithmetic progressions, zeros of Dirichlet L-functions

2010 Mathematics Subject Classification

Primary 11N13. Secondary 11M26, 11N69.

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