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 nonreal zeros of $\zeta (s)$, that the set of real numbers $x\ge 2$ for which $\pi (x)>\operatorname {li}(x)$ has a logarithmic density, which they computed to be about $2.6\times 10^{-7}$. 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 $\pi (p)>\operatorname {li}(p)$ 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 authors from a previous paper, leading to further progress on a 1988 conjecture of Erdős on primitive sets.