Primes in prime number races

TL;DR

Under RH and LI, the paper proves that the density of primes where $ ext{pi}(p) > ext{li}(p)$ matches the predicted density, extending to various prime number races and resolving longstanding conjectures.

Contribution

It establishes the existence and equality of the logarithmic density of primes in prime number races under RH and LI, including new results for Mertens and Zhang races.

Findings

The density of primes with $ ext{pi}(p) > ext{li}(p)$ exists and matches the Rubinstein-Sarnak density.

Results extend to Mertens and Zhang races, resolving previous open questions.

Progress on Erd ext{"o}s's 1988 conjecture on primitive sets.

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.