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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 2024 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
Proc. Amer. Math. Soc. 141 (2013), 409-420
DOI: https://doi.org/10.1090/S0002-9939-2012-11332-2
Published electronically: June 14, 2012

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.
References
  • J. Basquin, Sommes friables de fonctions multiplicatives aléatoires, Acta Arith., to appear.
  • Aline Bonami, Étude des coefficients de Fourier des fonctions de $L^{p}(G)$, Ann. Inst. Fourier (Grenoble) 20 (1970), no. fasc. 2, 335–402 (1971) (French, with English summary). MR 283496, DOI 10.5802/aif.357
  • S. Chatterjee and K. Soundararajan, Random multiplicative functions in short intervals, Int. Math. Res. Notices 2012, No. 3, 479–492.
  • Kai Lai Chung, A course in probability theory, Harcourt, Brace & World, Inc., New York, 1968. MR 0229268
  • Paul Erdős, Some unsolved problems, Magyar Tud. Akad. Mat. Kutató Int. Közl. 6 (1961), 221–254. MR 177846
  • A. Granville and K. Soundararajan, The distribution of values of $L(1,\chi _d)$, Geom. Funct. Anal. 13 (2003), no. 5, 992–1028. MR 2024414, DOI 10.1007/s00039-003-0438-3
  • G. Halász, On random multiplicative functions, Hubert Delange colloquium (Orsay, 1982) Publ. Math. Orsay, vol. 83, Univ. Paris XI, Orsay, 1983, pp. 74–96. MR 728404
  • A.J. Harper, Bounds on the suprema of Gaussian processes, and omega results for the sum of a random multiplicative function, preprint, arXiv:1012.0210.
  • Nicholas M. Katz and Peter Sarnak, Random matrices, Frobenius eigenvalues, and monodromy, American Mathematical Society Colloquium Publications, vol. 45, American Mathematical Society, Providence, RI, 1999. MR 1659828, DOI 10.1090/coll/045
  • P. Lévy, Sur les séries dont les termes sont des variables eventuelles indépendantes, Studia Math. 3 (1931), 119–155.
  • H. L. Montgomery, The pair correlation of zeros of the zeta function, Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972) Amer. Math. Soc., Providence, R.I., 1973, pp. 181–193. MR 0337821
  • Hugh L. Montgomery, Ten lectures on the interface between analytic number theory and harmonic analysis, CBMS Regional Conference Series in Mathematics, vol. 84, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1994. MR 1297543, DOI 10.1090/cbms/084
  • Daniel Revuz and Marc Yor, Continuous martingales and Brownian motion, 3rd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 293, Springer-Verlag, Berlin, 1999. MR 1725357, DOI 10.1007/978-3-662-06400-9
  • A. N. Shiryaev, Probability, 2nd ed., Graduate Texts in Mathematics, vol. 95, Springer-Verlag, New York, 1996. Translated from the first (1980) Russian edition by R. P. Boas. MR 1368405, DOI 10.1007/978-1-4757-2539-1
  • K. Soundararajan, Partial sums of the Möbius function, J. Reine Angew. Math. 631 (2009), 141–152. MR 2542220, DOI 10.1515/CRELLE.2009.044
  • G. Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres, Coll. Échelles, Berlin, 2008.
  • Aurel Wintner, Random factorizations and Riemann’s hypothesis, Duke Math. J. 11 (1944), 267–275. MR 10160
  • J. Wu, Note on a paper by A. Granville and K. Soundararajan: “The distribution of values of $L(1,\chi _d)$” [Geom. Funct. Anal. 13 (2003), no. 5, 992–1028; MR2024414], J. Number Theory 123 (2007), no. 2, 329–351. MR 2300818, DOI 10.1016/j.jnt.2006.07.004
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Bibliographic 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