Let $x$ and $y$ be two positive real numbers.
What is the mean value of the function $\log$ on $y$ friables integers less than $x$ i.e the value of the following sum
$$\sum\_{\substack{n \leq x \\ P(n)\leq y}} \log(n),$$
where $P(n)$ is the greatest prime factor of $n.$
Thanks in advance. 



Sum of log over friables
De La Breteche and Tenenbaum established in their paper **'Propriétés statistiques des entiers friables'** the asymptotic formula for the above sum. They find it as a corollary; Uniformly for $2 \leq y \leq x,$ we have 
$$\sum\_{\substack{n \leq x \\ P(n)\leq y}} \log(n)=\left\{\log(x)-\frac{\log(y)+O(\log\log(y))}{\log\left(1+\frac{y}{\log(x)}\right)} \right\}\Psi(x,y),$$
where $\Psi(x,y)$ is the number of friables integers less than $x$ whose greatest prime factors are less than $y.$


