Inclusive Prime Number Races

Let \( \pi(x;q,a) \) denote the number of primes up to \(x\) that are congruent to \(a( \text{mod }q) \). A "prime number race", for fixed modulus \(q\) and residue classes \(a_1, \ldots,a_r\), investigates the system of inequalities \[ \pi(x;q,a_1)> \pi(x;q,a_2)> \cdots > \pi(x;q,a_r). \] We expect that this system 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, the prime number race is called "inclusive". 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 with a substantially weaker linear independence hypothesis. This is joint work with Greg Martin.Non UBCUnreviewedAuthor affiliation: University of LethbridgeFacult