Sommes friables de fonctions multiplicatives al\'eatoires

TL;DR

This paper studies the behavior of random multiplicative functions on smooth integers, providing bounds for their summatory functions that improve previous estimates, especially when summing over all integers up to x.

Contribution

It introduces new upper bounds for the summatory functions of random multiplicative functions on friable integers, refining existing estimates and offering more precise formulas in various regions.

Findings

Bound $ ext{Psi}_f(x,y)$ by $ ext{Psi}(x,y)^{1/2+ ext{epsilon}}$

Improved estimate $M_f(x) ext{ll} ext{sqrt}(x) ( ext{log} ext{log} x)^{2+ ext{epsilon}}$

Enhanced understanding of random multiplicative functions on smooth integers

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.