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Abstract

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 $P(n)\leq y$ and $n\leq x$. 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 $(\log x^c) \leq y$ 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.