A generalized Kubilius-Barban-Vinogradov bound for prime multiplicities

Abstract

We present an assessment of the distance in total variation of arbitrary collections of prime factor multiplicities of a random number in [n] = {1,...,n} and a collection of independent geometric random variables. More precisely, we impose mild conditions on the probability law of the random sample and the aforementioned collection of prime multiplicities, for which a fast decaying bound on the distance towards a tuple of geometric variables holds. Our results generalize and complement those from Kubilius (1964) and Barban and Vinogradov (1964) which consider the particular case of uniform samples in [n] and collection of “small primes”. As applications, we show a generalized version of the celebrated Erdös Kac theorem for not necessarily uniform samples of numbers.