Friable integers: Turán-Kubilius inequality and applications. (Entiers friables: inégalité de Turán-Kubilius et applications.) (French) Zbl 1182.11045

de la Bretèche, R.; Tenenbaum, G.

Invent. Math. 159, No. 3, 531-588 (2005).

Let \(P(n)\) be the largest prime divisor of natural number \(n>1\), \(P(1)=1\), and \(S(x,y)=\left\{n\leq x: P(n)\leq y \right\}\) be the set of \(y\)-friable positive integers not exceeding \(x\).
A bouquet of known classical problems of number theory is solved for friable positive integers. The analogue of the Turán-Kubilius inequality is proved on \(S(x,y)\) for additive complex function. The asymptotic formulas are obtained for the best possible constant in the Turán-Kubilius inequality. The analogue of the Erdős-Wintner theorem is presented. The generalization of the Daboussi theorem [see H. Daboussi, Prog. Math. 85, 111–118 (1990; Zbl 0719.11059)] for exponential sums is obtained. Finally, an interesting result on the prime divisors of friable integers is proved.
The Turán-Kubilius inequality on \(S(x,y)\) is the main result of the present paper, because the other results are obtained essentially using this inequality. It should be noted that one of the obtained asymptotic formulas for the constant in the Turán-Kubilius inequality revises A. Hildebrand’s results [see Math. Z. 183, 145–170 (1983; Zbl 0493.10049)].

Reviewer: Jonas Šiaulys (Vilnius)

MSC:

11N64	Other results on the distribution of values or the characterization of arithmetic functions
11N37	Asymptotic results on arithmetic functions
11L07	Estimates on exponential sums

Keywords:

Turán-Kubilius inequality; smooth integers; Erdoes-Wintner theorem; variance of additive functions

Citations:

Zbl 0719.11059; Zbl 0493.10049

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