Large prime factors of well-distributed sequences

2024/02/19 by Bharadwaj, Abhishek, Rodgers, Brad · 1 citation
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2402.11884

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-ε, showing that this probability is O(ε). 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.

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