Inclusive prime number races

Abstract

Let $\pi(x;q,a)$ denote the number of primes up to $x$ that are congruent to $a$ modulo $q$. A prime number race, for fixed modulus $q$ and residue classes $a_1, \ldots, a_r$, investigates the system of inequalities $\pi(x;q,a_1) > \pi(x;q,a_2) > \cdots > \pi(x;q,a_r)$. The study of prime number races was initiated by Chebyshev and further studied by many others, including Littlewood, Shanks-R\'{e}nyi, Knapowski-Turan, and Kaczorowski. We expect that this system of inequalities should have arbitrarily large solutions $x$, and moreover we expect the same to be true no matter how we permute the residue classes $a_j$; if this is the case, and if the logarithmic density of the set of such $x$ exists and is positive, the prime number race is called inclusive. In breakthrough research, Rubinstein and Sarnak proved conditionally that every prime number race is inclusive; they assumed not only the generalized Riemann hypothesis but also a strong statement about the linear independence of the zeros of Dirichlet $L$-functions. We show that the same conclusion can be reached assuming the generalized Riemann hypothesis and a substantially weaker linear independence hypothesis. In fact, we can assume that almost all of the zeros may be involved in $\mathbb{Q}$-linear relations; and we can also conclude more strongly that the associated limiting distribution has mass everywhere. This work makes use of a number of ideas from probability, the explicit formula from number theory, and the Kronecker-Weyl equidistribution theorem.