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 ζ ( s ) \zeta (s) , that the set of real numbers x ≥ 2 x\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 − 7 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 p 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 e γ log ⁡ x e^{\gamma }\log x and the “Zhang race” between ∑ p ≥ x 1 / ( p log ⁡ p ) \sum _{p\ge x}1/(p\log p) and 1 / log ⁡ x 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.