An integer n is said to be y-friable if its greatest prime factor P(n) is less than y. In this paper, we study numbers of the shape n-1 when P(n) ≤ y and n ≤ x. One expects that, statistically, their multiplicative behaviour resembles that of all integers less than x. Extending a result of Basquin (2010), we estimate the mean value over shifted friable numbers of certain arithmetic functions when for some positive c, showing a change in behaviour according to whether log y/log x tends to infinity or not. In the same range in (x, y), we prove an Erdős-Kac-type theorem for shifted friable numbers, improving a result of Fouvry Tenenbaum (1996). The results presented here are obtained using recent work of Harper (2012) on the statistical distribution of friable numbers in arithmetic progressions.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm137-2-1,
author = {Sary Drappeau},
title = {Propri\'et\'es multiplicatives des entiers friables translat\'es},
journal = {Colloquium Mathematicae},
volume = {135},
year = {2014},
pages = {149-164},
language = {fra},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm137-2-1}
}
Sary Drappeau. Propriétés multiplicatives des entiers friables translatés. Colloquium Mathematicae, Tome 135 (2014) pp. 149-164. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm137-2-1/