The Density of Borromean Primes

Abstract

In this paper, we study an asymptotic distribution of sets of primes satisfying certain "linking conditions" in arithmetic topology, namely, conditions given by the Legendre and R\'edei symbols among sets of primes. As our Main Theorem, we prove an asymptotic density formula for Borromean primes among all primes. For the proof, we use the effective Chebotarev density formula under the Generalized Riemann Hypothesis and explicit computations of discriminants of the number fields involved in R\'edei's extension.