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 investigates the system of inequalities $\pi (x;q,a_1) > \pi (x;q,a_2) > \cdots > \pi (x;q,a_r)$ for fixed modulus $q$ and residue classes $a_1, \ldots , a_r$. The study of prime number races was initiated by Chebyshev and further studied by many others, including Littlewood, Shanks–Rényi, 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 (1994) 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.