A generalized Kubilius-Barban-Vinogradov bound for prime multiplicities

Louis H. Y. Chen; Arturo Jaramillo; Xiaochuan Yang

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arXiv · 2111.07361

A generalized Kubilius-Barban-Vinogradov bound for prime multiplicities

Louis H. Y. Chen·Arturo Jaramillo·Xiaochuan Yang

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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.

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Louis H. Y. Chen, Arturo Jaramillo, Xiaochuan Yang. 2021-11-14. A generalized Kubilius-Barban-Vinogradov bound for prime multiplicities. https://arxiv.org/abs/2111.07361

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