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Facteurs premiers de sommes d'entiers


Author: Gérald Tenenbaum
Journal: Proc. Amer. Math. Soc. 106 (1989), 287-296
MSC: Primary 11N60; Secondary 11B75
DOI: https://doi.org/10.1090/S0002-9939-1989-0952323-0
MathSciNet review: 952323
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Abstract: Let $ \Omega(n)$ denote the total number of prime factors of the positive integer $ n$, and set for real $ x$, $ \alpha$, and integer $ k \geq 1$,

$\displaystyle \pi_k(x) = \sum\limits_{\substack{n \leqslant x \\ \Omega (n) = k... ...n), \quad E(x,\alpha) = x^{-1}\sum\limits_{n \leqslant x} \mathbf{e}(\alpha n),$

where $ {\mathbf{e}}(t): = \exp (2\pi it)$. We establish a best possible "independence" result of the type

$\displaystyle {\pi _k}(x,\alpha )/{\pi _k}(x) = E(x,\alpha ) + O({\delta _k}(x))$

which is valid uniformly in $ x,k,\alpha $, and where the error $ {\delta _k}(x)$ tends to 0 as $ x \to + \infty $, if, and only if, $ k \sim \operatorname{log} \operatorname{log} x$. As an application we prove a recent conjecture of Erdös, Maier, and Sárközy concerning the remainder in their Erdös-Kac theorem for sum-sets.

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DOI: https://doi.org/10.1090/S0002-9939-1989-0952323-0
Article copyright: © Copyright 1989 American Mathematical Society

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