Distributional Properties of the Largest Prime Factor

Abstract

Let P(n) denote the largest prime factor of an integer n ≥ 2, and put P(1) = 1. In this paper, we study the distribution of the sequence {P(n) : n ≥ 1} over the set of congruence classes modulo an integer q ≥ 2, and we study the same question for the sequence {P(p − 1) : p is prime}. We also give bounds for rational exponential sums involving P(n). Finally, for an irrational number _, we show that the sequence {_P(n) : n ≥ 1} is uniformly distributed modulo 1.