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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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


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
  • Email:
  • Gérald Tenenbaum
  • Affiliation: Institut Élie Cartan Nancy, Nancy-Université, CNRS & INRIA, 54506 Vandœuvre-lès-Nancy, France
  • ORCID: 0000-0002-0478-3693
  • Email:
  • Jie Wu
  • Affiliation: Institut Élie Cartan Nancy, Nancy-Université, CNRS & INRIA, 54506 Vandœuvre-lès-Nancy, France
  • Email:
  • 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
  • © Copyright 2012 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 141 (2013), 409-420
  • MSC (2010): Primary 11N37; Secondary 11K99, 60F15
  • DOI:
  • MathSciNet review: 2996946