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On mean values of random multiplicative functions

Authors: Yuk-Kam Lau, Gérald Tenenbaum and Jie Wu
Journal: Proc. Amer. Math. Soc. 141 (2013), 409-420
MSC (2010): Primary 11N37; Secondary 11K99, 60F15
Published electronically: June 14, 2012
MathSciNet review: 2996946
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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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Additional Information

Yuk-Kam Lau
Affiliation: Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong

Gérald Tenenbaum
Affiliation: Institut Élie Cartan Nancy, Nancy-Université, CNRS & INRIA, 54506 Vandœuvre-lès-Nancy, France
ORCID: 0000-0002-0478-3693

Jie Wu
Affiliation: Institut Élie Cartan Nancy, Nancy-Université, CNRS & INRIA, 54506 Vandœuvre-lès-Nancy, France

Keywords: Random multiplicative functions, law of iterated logarithm, Riemann hypothesis, Möbius function, mean values of multiplicative functions, Rademacher functions.
Received by editor(s): December 4, 2010
Received by editor(s) in revised form: December 5, 2010, and June 30, 2011
Published electronically: June 14, 2012
Communicated by: Richard C. Bradley
Article copyright: © Copyright 2012 American Mathematical Society