A generalized Kubilius-Barban-Vinogradov bound for prime multiplicities. (English) Zbl 1532.60006
Summary: We present an assessment of the distance in total variation of arbitrary collections of prime factor multiplicities of a random number in \([n] = \{1, \ldots, n\}\) 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 J. Kubilius [Probabilistic methods in the theory of numbers. Providence, RI: American Mathematical Society (AMS) (1964; Zbl 0133.30203)] and M. B. Barban and A. I. Vinogradov [Sov. Math., Dokl. 5, 96–98 (1964; Zbl 0166.05702); translation from Dokl. Akad. Nauk SSSR 154, 495–496 (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 Erdős Kac theorem for not necessarily uniform samples of numbers.
MSC:
| 60B12 | Limit theorems for vector-valued random variables (infinite-dimensional case) |
| 11K65 | Arithmetic functions in probabilistic number theory |
References:
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