Inclusive prime number races

TL;DR

This paper proves that prime number races are inclusive under the generalized Riemann hypothesis and a weaker linear independence assumption, showing the distribution of primes in residue classes is more evenly spread than previously established.

Contribution

It weakens the linear independence hypothesis needed to prove the inclusiveness of prime number races, allowing for more zeros involved in rational relations.

Findings

Prime number races are inclusive under weaker hypotheses.

The limiting distribution of prime races has full support.

Almost all zeros can be involved in rational relations.

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.