Primes in prime number races

Rubinstein and Sarnak have shown, conditional on the Riemann hypothesis (RH) and the linear independence hypothesis (LI) on the nonreal zeros of ζ(s)\zeta (s), that the set of real numbers x≥2x\ge 2 for which π(x)>li⁡(x)\pi (x)>\operatorname {li}(x) has a logarithmic density, which they computed to be about 2.6×10−72.6\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 pp for which π(p)>li⁡(p)\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 ∏p>x(1−1/p)−1\prod _{p> x}(1-1/p)^{-1} and eγlog⁡xe^{\gamma }\log x and the “Zhang race” between ∑p≥x1/(plog⁡p)\sum _{p\ge x}1/(p\log p) and 1/log⁡x1/\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.