Sommes friables de fonctions multiplicatives al\'eatoires

We consider a sequence $\{f(p){\}}_{p\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{m}\mathrm{e}}$ of independent random variables taking values $\pm 1$ with probability $1\mathrm{/}2$, and extend $f$ to a multiplicative arithmetic function defined on the squarefree integers. We investigate upper bounds for ${\mathrm{\Psi }}_f(x,y)$, the summatory function of $f$ on $y$-friable integers $\le x$. We obtain estimations of the type $\Psi_f(x,y) \ll \Psi(x,y)^{1/2+\epsilon}$, more precise formulas being given in suitable regions for $x,y$. In the special case $y=x$, this leads to the estimate $M_f(x)={\sum }_{n\le x}f(n)\ll \sqrt{x}(\log \log x{)}^{2+ϵ}$, which improves on previous bounds.