Sommes friables de fonctions multiplicatives al\'eatoires

Abstract

We consider a sequence $\{f(p)\}_{p\ {\rm prime}}$ of independent random variables taking values $\pm 1$ with probability $1/2$, and extend $f$ to a multiplicative arithmetic function defined on the squarefree integers. We investigate upper bounds for $\Psi_f(x,y)$, the summatory function of $f$ on $y$-friable integers $\leq x$. 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 $M_f(x) = \sum_{n \leq x} f(n) \ll \sqrt{x}\, (\log \log x)^{2+\epsilon}$, which improves on previous bounds.