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Joseph Basquin


17/01/2021

On random multiplicative functions

We consider a sequence $(f(p))_{p \ prime}$ of independent random variables taking values ±1 with probability 1/2, and extend $f$ to a multiplicative arithmetic function defined on the squarefree integers.

Finding an upper bound for $M(x) = \sum_{n \leq x} f(x)$ has been long studied. Wintner proved in 1944 that $M(x) \ll x^{1/2 + \varepsilon}$ a.e., later improved by Erdös who establishes $M(x) \ll \sqrt{x} (\log x)^c$. Halász then obtains in 1983 the upper bound $M(x) \ll \sqrt{x} e^{c \sqrt{(\log\log x)(\log\log \log x)}}$ a.e. In a preliminary work by Lau, Tenenbaum, Wu (improved since then), the bound $M(x) \ll \sqrt{x} (\log \log x)^{5/2 + \varepsilon}$ has been obtained.

The use of martingale methods, a generalization of the Doob inequality (Hájek-Renyi inequality) and other techniques allows to obtain

$$M(x) \ll \sqrt{x} (\log \log x)^{2 + \varepsilon} \qquad\textrm{a.e.}$$

This was the goal of my work Sommes friables de fonctions multiplicatives aléatoires published in Acta Arith., 2012, as well as obtaining estimations of the type $\Psi_f(x,y)\ll \Psi(x,y)^{1/2+\varepsilon}$ on y-smooth (a.k.a. friable) integers ≤ x.

Is it possible to improve the exponent $2+\varepsilon$ further? The question remains open (the exponent $3/2 + \varepsilon$ had been claimed – but then removed in an updated version – in a more recent article).