Prime number races with three or more competitors

Abstract

Let \(q\ge 3\) and \(2\le r\le \phi (q)\) be positive integers, and \(a_1,\ldots ,a_r\) be distinct reduced residue classes modulo \(q\). Rubinstein and Sarnak defined \(\delta (q;a_1,\ldots ,a_r)\) to be the logarithmic density of the set of real numbers \(x\) such that \(\pi (x;q,a_1)>\pi (x;q,a_2)>\cdots >\pi (x;q,a_r)\). In this paper, we establish an asymptotic formula for \(\delta (q;a_1,\ldots ,a_r)\) when \(r\ge 3\) is fixed and \(q\) is large. Several applications concerning these prime number races are then deduced. First, comparing with a recent work of Fiorilli and Martin on the case \(r=2\), we show that these densities behave differently when \(r\ge 3\). Another surprising consequence of our results is that, unlike two-way races, biases do appear in races involving three or more squares (or non-squares) to large moduli. Furthermore, we establish a partial result towards a conjecture of Rubinstein and Sarnak on biased races, and disprove a recent conjecture of Feuerverger and Martin concerning bias factors. Lastly, we use our method to derive the Fiorilli and Martin asymptotic formula for the densities when \(r=2\).