paper

Densities in certain three-way prime number races

arXiv:1908.10902 · doi:10.4153/S0008414X20000747

Abstract

Let $a_{1}$ , $a_{2}$ , and $a_{3}$ be distinct reduced residues modulo $q$ satisfying the congruences $a_{1}^{2} \equiv a_{2}^{2} \equiv a_{3}^{2} (mod q)$ . We conditionally derive an asymptotic formula, with an error term that has a power savings in $q$ , for the logarithmic density of the set of real numbers $x$ for which $I ¨ (x; q, a_{1}) > I ¨ (x; q, a_{2}) > I ¨ (x; q, a_{3})$ . The relationship among the $a_{i}$ allows us to normalize the error terms for the $I ¨ (x; q, a_{i})$ in an atypical way that creates mutual independence among their distributions, and also allows for a proof technique that uses only elementary tools from probability.

31 pages