Primes in prime number races

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 $x\ge 2$ for which $\pi (x)>$ 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 $\pi (p)>$ 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$e^{\gamma}\log x$andthe\mathrm{"}Zhangrace\mathrm{"}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.