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Proceedings of the American Mathematical Society

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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

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
  • Email: yklau@maths.hku.hk
  • Gérald Tenenbaum
  • Affiliation: Institut Élie Cartan Nancy, Nancy-Université, CNRS & INRIA, 54506 Vandœuvre-lès-Nancy, France
  • ORCID: 0000-0002-0478-3693
  • Email: gerald.tenenbaum@iecn.u-nancy.fr
  • Jie Wu
  • Affiliation: Institut Élie Cartan Nancy, Nancy-Université, CNRS & INRIA, 54506 Vandœuvre-lès-Nancy, France
  • Email: wujie@iecn.u-nancy.fr
  • 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: https://doi.org/10.1090/S0002-9939-2012-11332-2
  • MathSciNet review: 2996946