Primes in prime number races

Abstract

Rubinstein and Sarnak have shown, conditional on the Riemann hypothesis (RH) and the linear independence hypothesis (LI) on the non-real zeros of $\zeta(s)$, that the set of real numbers $x\ge2$ for which $\pi(x)>$ li$(x)$ has a logarithmic density, which they computed to be about $2.6\times10^{-7}$. A natural problem is to examine the actual primes in this race. We prove, assuming RH and LI, that the logarithmic density of the set of primes $p$ for which $\pi(p)>$ li$(p)$ relative to the prime numbers exists and is the same as the Rubinstein-Sarnak density. We also extend such results to a broad class of prime number races, including the "Mertens race" between $\prod_{p< x}(1-1/p)^{-1}$ and $e^{\gamma}\log x$ and the "Zhang race" between $\sum_{p\ge x}1/(p\log p)$ and $1/\log x$. These latter results resolve a question of the first and third author from a previous paper, leading to further progress on a 1988 conjecture of Erd\H{o}s on primitive sets.