Sommes friables de fonctions multiplicatives al\'eatoires

摘要

We consider a sequence $\{f(p)\}_{p\ {\rm prime}}$ of independent randomvariables taking values $\pm 1$ with probability $1/2$, and extend $f$ to amultiplicative arithmetic function defined on the squarefree integers. Weinvestigate 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 suitableregions 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.