Large prime factors of well-distributed sequences

Abstract

We study the distribution of large prime factors of a random element $u$ of arithmetic sequences satisfying simple regularity and equidistribution properties. We show that if such an arithmetic sequence has level of distribution $1$ the large prime factors of $u$ tend to a Poisson-Dirichlet process, while if the sequence has any positive level of distribution the correlation functions of large prime factors tend to a Poisson-Dirichlet process against test functions of restricted support. For sequences with positive level of distribution, we also estimate the probability the largest prime factor of $u$ is greater than $u^{1-\epsilon}$, showing that this probability is $O(\epsilon)$. Examples of sequences described include shifted primes and values of single-variable irreducible polynomials. The proofs involve (i) a characterization of the Poisson-Dirichlet process due to Arratia-Kochman-Miller and (ii) an upper bound sieve.