Prime number races with three or more competitors

Abstract

Fix an integer $r\geq 3$. Let $q$ be a large positive integer and $a_1,...,a_r$ be distinct residue classes modulo $q$ that are relatively prime to $q$. In this paper, we establish an asymptotic formula for the logarithmic density $\delta_{q;a_1,...,a_r}$ of the set of real numbers $x$ such that $\pi(x;q,a_1)>\pi(x;q,a_2)>...>\pi(x;q,a_r),$ as $q\to\infty$; conditionally on the assumption of the Generalized Riemann Hypothesis GRH and the Grand Simplicity Hypothesis GSH. Several applications concerning these prime number races are then deduced. Indeed, comparing with a recent work of D. Fiorilli and G. Martin for the case $r=2$, we show that these densities behave differently when $r\geq 3$. Another consequence of our results is the fact that, unlike two-way races, biases do appear in races involving three of more squares (or non-squares) to large moduli. Furthermore, we establish a conjecture of M. Rubinstein and P. Sarnak (on biased races) in certain cases where the $a_i$ are assumed to be fixed and $q$ is large. We also prove that a conjecture of A. Feuerverger and G. Martin concerning "bias factors" (which follows from the work of Rubinstein and Sarnak for $r=2$) does not hold when $r\geq 3$. Finally, we use a variant of our method to derive Fiorilli and Martin asymptotic formula for the densities in two-way races.