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On mean values of random multiplicative functions. (English) Zbl 1294.11167

For \(p\) prime, let \(f(p)\) be a sequence of independent random variables satisfying \(\mathbb{P}(f(p)=1) = \mathbb{P}(f(p)=-1) = 1/2\). We then define a random multiplicative function \(f(n)\) by setting \(f(n) := \prod_{p\mid n}f(p)\) if \(n\) is squarefree, and \(f(n) := 0\) if \(n\) is not squarefree. Thus \(f(n)\) is a heuristic model for the Möbius function \(\mu(n)\), or for real Dirichlet characters, as well as simply being an interesting probabilistic object. Actually this paper considers a slightly more general model where \(f(p)\) is sometimes allowed to equal zero, but we shall ignore that for the purposes of this review.
In this paper, the authors show that for any fixed \(\varepsilon > 0\) we almost surely (over all realisations of the \(f(p)\)) have \[ \sum_{n \leq x} f(n) \ll_{f,\varepsilon} \sqrt{x} (\log\log x)^{2+\varepsilon} \qquad \text{as} \; x \rightarrow \infty . \] Here the implicit constant may depend on \(\varepsilon\) and on the realisation of \(f\), but is independent of \(x\). It should be noted that the published version of the paper actually claims a bound \(\ll_{f,\varepsilon} \sqrt{x} (\log\log x)^{3/2+\varepsilon}\), but there is a small error on page 417 (a term \(r=1\) is omitted) that invalidates this, and a corrected version with the exponent \(2+\varepsilon\) is available on the second author’s webpage.
In any case, the bound improves substantially on the previous best result of G. Halász [Publ. Math. Orsay 83.04, 74–96 (1983; Zbl 0522.10033)], which was an almost sure bound \(\ll_{f} \sqrt{x} e^{c\sqrt{\log\log x \log\log\log x}}\).
It seems worthwhile to compare the authors’ proof with the approach one would take for sums \(\sum_{n \leq x} \varepsilon_{n}\) of completely independent random signs, as in the Law of the Iterated Logarithm. In that case, one would choose a sequence of break points \(x_{l}\) that are quite widely spaced, e.g \(x_{l} = (1+\delta)^{l}\). Using strong exponential concentration inequalities, one has that \[ \mathbb{P}\left(\sum_{n \leq x_{l}} \varepsilon_{n} \geq \sqrt{(1+\delta)2x_{l} \log\log x_{l}}\right) \ll 1/\log^{1+\delta} x_{l} \ll_{\delta} 1/l^{1+\delta} . \] Summing over \(l\) one obtains a convergent series, so the Borel-Cantelli lemma implies that almost surely \(|\sum_{n \leq x_{l}} \varepsilon_{n}| \leq \sqrt{(1+\delta)2x_{l} \log\log x_{l}}\) for sufficiently large \(l\). Finally, one can argue that the sums don’t vary much between different points \(x_{l}\), and by letting \(\delta\) tend to zero one obtains a sharp upper bound.
The key problem for the authors is that there are strong dependencies between the values \(f(n)\), (e.g. \(f(6)\) is completely determined by \(f(2)\) and \(f(3)\)), so they don’t have access to exponential concentration results. Indeed, the crucial term in their final bound comes simply from an application of a second moment inequality (see display (3.14) in the paper). Thus if one attempted to use the authors’ methods to understand \(\sum_{n \leq x} f(n)\) for any fixed \(x\) one would obtain fairly weak results. But they show that, by conditioning in turn on the behaviour of \(f(p)\) for various blocks of primes, one can control the behaviour of all the sums \(\sum_{n \leq x} f(n)\) in terms of the behaviour of certain integrals at extremely widely spaced sample points \(X_{l} = e^{2^{l}}\) (cf. their Lemma 3.1). Thus they only have to sum their bounds over a very sparse sequence to apply Borel-Cantelli. It should be mentioned that the authors’ main innovation compared with Halász is in the way they perform their conditioning: they divide the sequence of primes into blocks so that a typical integer will have very few prime factors in each block, which makes the arguments much more efficient.
The true size of the fluctuations of \(\sum_{n \leq x} f(n)\) remains mysterious and tantalising, since the best known lower bound (due to the reviewer [Ann. Appl. Prob. 23, No. 2, 584–616 (2013; Zbl 1268.60075)]) asserts that any bound \(\ll_{f,\varepsilon} \sqrt{x} (\log\log x)^{-5/2-\varepsilon}\) is almost surely false. Thus, for all we know, the sum of a random multiplicative function might often have better than square-root cancellation, or it might behave just like a sum of independent random signs, or it might exhibit larger fluctuations than that, up to the authors’ upper bound.
Finally, let us note that the paper under review is very nicely written, and is recommended reading for anybody interested in random multiplicative functions.

MSC:

11N37 Asymptotic results on arithmetic functions
11K99 Probabilistic theory: distribution modulo \(1\); metric theory of algorithms
60F15 Strong limit theorems
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References:

[1] J. Basquin, Sommes friables de fonctions multiplicatives aléatoires, Acta Arith., to appear.
[2] Aline Bonami, Étude des coefficients de Fourier des fonctions de \?^{\?}(\?), Ann. Inst. Fourier (Grenoble) 20 (1970), no. fasc. 2, 335 – 402 (1971) (French, with English summary). · Zbl 0195.42501
[3] S. Chatterjee and K. Soundararajan, Random multiplicative functions in short intervals, Int. Math. Res. Notices 2012, No. 3, 479-492. · Zbl 1248.11056
[4] Kai Lai Chung, A course in probability theory, Harcourt, Brace & World, Inc., New York, 1968. · Zbl 0980.60001
[5] Paul Erdős, Some unsolved problems, Magyar Tud. Akad. Mat. Kutató Int. Közl. 6 (1961), 221 – 254. · Zbl 0100.02001
[6] A. Granville and K. Soundararajan, The distribution of values of \?(1,\?_{\?}), Geom. Funct. Anal. 13 (2003), no. 5, 992 – 1028. · Zbl 1044.11080 · doi:10.1007/s00039-003-0438-3
[7] G. Halász, On random multiplicative functions, Hubert Delange colloquium (Orsay, 1982) Publ. Math. Orsay, vol. 83, Univ. Paris XI, Orsay, 1983, pp. 74 – 96.
[8] 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. · Zbl 1268.60075
[9] 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. · Zbl 0958.11004
[10] P. Lévy, Sur les séries dont les termes sont des variables eventuelles indépendantes, Studia Math. 3 (1931), 119-155.
[11] 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.
[12] 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. · Zbl 0814.11001
[13] 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. · Zbl 0917.60006
[14] 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. · Zbl 0508.60001
[15] K. Soundararajan, Partial sums of the Möbius function, J. Reine Angew. Math. 631 (2009), 141 – 152. · Zbl 1184.11040 · doi:10.1515/CRELLE.2009.044
[16] G. Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres, Coll. Échelles, Berlin, 2008. · Zbl 0788.11001
[17] Aurel Wintner, Random factorizations and Riemann’s hypothesis, Duke Math. J. 11 (1944), 267 – 275. · Zbl 0060.10510
[18] J. Wu, Note on a paper by A. Granville and K. Soundararajan: ”The distribution of values of \?(1,\?_{\?})” [Geom. Funct. Anal. 13 (2003), no. 5, 992 – 1028; MR2024414], J. Number Theory 123 (2007), no. 2, 329 – 351. · Zbl 1197.11107 · doi:10.1016/j.jnt.2006.07.004
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