A generalized Kubilius-Barban-Vinogradov bound for prime multiplicities

TL;DR

This paper generalizes bounds on the total variation distance between prime factor multiplicities of random numbers and independent geometric variables, extending previous results and applying to generalized Erdős–Kac theorems.

Contribution

It introduces a generalized bound for prime multiplicities in arbitrary samples, broadening the scope beyond uniform distributions and small primes.

Findings

Derived a fast decaying bound on total variation distance

Extended Erdős–Kac theorem to non-uniform samples

Generalized previous bounds for prime factorization distributions

Abstract

We present an assessment of the distance in total variation of \textit{arbitrary} collection of prime factor multiplicities of a random number in $[n]=\{1,\dots, n\}$ 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 et al. 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 Erd\"os Kac theorem for not necessarily uniform samples of numbers.