An almost sure upper bound for random multiplicative functions on integers with a large prime factor. (English) Zbl 1483.11167
Summary: Let \(f\) be a Rademacher or a Steinhaus random multiplicative function. Let \(\varepsilon > 0\) small. We prove that, as \(x\to +\infty\), we almost surely have
\[
\Big|\sum \limits_{\substack{n\le x \\ P(n)>\sqrt{x}}}f(n)\Big|\le \sqrt{x}{(\log \log x)^{1/ 4+\varepsilon}},
\] where \(P(n)\) stands for the largest prime factor of \(n\). This gives an indication of the almost sure size of the largest fluctuations of \(f\).
MSC:
| 11K65 | Arithmetic functions in probabilistic number theory |
| 11N64 | Other results on the distribution of values or the characterization of arithmetic functions |
Keywords:
random multiplicative functions; law of iterated logarithm; Borel-Cantelli lemma; sums of independent random variables; low momentsReferences:
| [1] | J. Basquin. Sommes friables de fonctions multiplicatives aléatoires. Acta Arithmetica, 152.3 (2012). · Zbl 1304.11103 |
| [2] | A. Bonami. Étude des coefficients de Fourier des fonctions de \[{L_p}(G)\]. Ann. Inst. Fourier, Grenoble 20, (2) 335-402 (1970). · Zbl 0195.42501 |
| [3] | P. Erdős. Some applications of probability methods to number theory. In Proc. of the 4th Pannonian Symp. on Math. Stat., (Bad Tatzmannsdorf, Austria 1983). Mathematical statistics and applications, vol. B, pp 1-18. Reidel, Dordrecht (1985). · Zbl 0593.10039 |
| [4] | A. Gut. Probability: a graduate course. New York: Springer (2005). · Zbl 1076.60001 |
| [5] | G. Halász. On random multiplicative functions. In Hubert Delange Colloquium, (Orsay, 1982). Publications Mathématiques d’Orsay, 83, pp 74-96. Univ. Paris XI, Orsay (1983). · Zbl 0522.10033 |
| [6] | A. J. Harper. Bounds on the suprema of Gaussian processes, and omega results for the sum of a random multiplicative function. Ann. Appl. Probab., 23, no. 2, pp 584-616 (2013). · Zbl 1268.60075 |
| [7] | A. J. Harper. Moments of random multiplicative functions, II: High moments. Algebra and Number Theory, Vol. 13, No. 10, 2277-2321 (2019). · Zbl 1472.11265 |
| [8] | A. J. Harper. Moments of random multiplicative functions, I: Low moments, better than squareroot cancellation, and critical multiplicative chaos. Forum of Mathematics, Pi, 8, e1, 95pp. (2020). · Zbl 1472.11254 |
| [9] | A. J. Harper. Almost sure large fluctuations of random multiplicative functions. Preprint https://arxiv.org/abs/2012.15809. |
| [10] | A. J. Harper, A. Nikeghbali, M. Radziwiłł. A note on Helson’s conjecture on moments of random multiplicative functions. Analytic number theory, pp 145-169, Springer, Cham. (2015). · Zbl 1391.11140 |
| [11] | W. Hoeffding. Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58, 13-30 (1963). · Zbl 0127.10602 |
| [12] | Y.-K. Lau, G. Tenenbaum, J. Wu. On mean values of random multiplicative functions. Proceedings of the American Mathematical Society. Volume 141, Number 2, pp. 409-420 (2013). Also see https://tenenb.perso.math.cnrs.fr/PPP/RMF.pdf for some corrections to the published version. · Zbl 1294.11167 |
| [13] | H. Montgomery, R. Vaughan. Multiplicative Number theory I: Classical theory. Cambridge U. P. (2006). |
| [14] | P. Shiu. A Brun-Titchmarsh theorem for multiplicative functions. J. Reine Angew. Math. 313, 161-170 (1980). · Zbl 0412.10030 |
| [15] | K. Soundararajan. Partial sums of the Möbius function. J. reine angew. Math. 631, 141-152 (2009). · Zbl 1184.11040 |
| [16] | G. Tenenbaum. Introduction to Analytic and Probabilistic Number Theory. Third edition, Graduate Studies in Mathematics, 163, American Mathematical Society, Providence, RI (2015). · Zbl 1336.11001 |
| [17] | A. Wintner. Random factorizations and Riemann’s hypothesis. Duke Math. J., 11, pp 267-275 (1944). · Zbl 0060.10510 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
