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 that 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.