On mean values of random multiplicative functions

Lau, Yuk-Kam; Tenenbaum, Gérald; Wu, Jie

doi:10.1090/S0002-9939-2012-11332-2

On mean values of random multiplicative functions
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by Yuk-Kam Lau, Gérald Tenenbaum and Jie Wu PDF

Proc. Amer. Math. Soc. 141 (2013), 409-420 Request permission

Abstract:

Let $\mathscr P$ denote the set of primes and $\{f(p)\}_{p\in \mathscr P}$ be a sequence of independent Bernoulli random variables taking values $\pm 1$ with probability $1/2$. Extending $f$ by multiplicativity to a random multiplicative function $f$ supported on the set of squarefree integers, we prove that, for any $\varepsilon >0$, the estimate $\sum _{n\leqslant x}f(n)\ll \sqrt {x} (\log \log x)^{3/2+\varepsilon }$ holds almost surely, thus qualitatively matching the law of the iterated logarithm, valid for independent variables. This improves on corresponding results by Wintner, Erdős and Halász.

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