# zbMATH — the first resource for mathematics

Partial sums of biased random multiplicative functions. (English) Zbl 1414.11115
Summary: Let $$\mathcal{P}$$ be the set of the primes. We consider a class of random multiplicative functions $$f$$ supported on the squarefree integers, such that $$\{f(p) \}_{p \in \mathcal{P}}$$ form a sequence of $$\pm$$1 valued independent random variables with $$\mathbb{E} f(p) < 0$$, $$\text{for all } p \in \mathcal{P}$$. The function $$f$$ is called strongly biased (towards classical Möbius function), if $$\sum_{p \in \mathcal{P}} \frac{f(p)}{p} = - \infty$$ a.s., and it is weakly biased if $$\sum_{p \in \mathcal{P}} \frac{f(p)}{p}$$ converges a.s. Let $$M_f(x) : = \sum_{n \leq x} f(n)$$. We establish a number of necessary and sufficient conditions for $$M_f(x) = o(x^{1 - \alpha})$$ for some $$\alpha > 0$$, a.s., when $$f$$ is strongly or weakly biased, and prove that the Riemann Hypothesis holds if and only if $$M_{f_\alpha}(x) = o(x^{1 / 2 + \varepsilon})$$ for all $$\varepsilon > 0$$ a.s., for each $$\alpha > 0$$, where $$\{f_\alpha \}_\alpha$$ is a certain family of weakly biased random multiplicative functions.
##### MSC:
 11N37 Asymptotic results on arithmetic functions 11A25 Arithmetic functions; related numbers; inversion formulas 11K65 Arithmetic functions in probabilistic number theory
Full Text:
##### References:
  Apostol, T. M., Introduction to analytic number theory, Undergrad. Texts Math., (1976), Springer-Verlag New York · Zbl 0335.10001  Basquin, J., Sommes friables de fonctions multiplicatives aléatoires, Acta Arith., 152, 243-266, (2012) · Zbl 1304.11103  Bradley, R. C., Every “lower psi-mixing“ Markov chain is “interlaced rho-mixing”, Stochastic Process. Appl., 72, 221-239, (1997) · Zbl 0948.60064  Bradley, R. C., Basic properties of strong mixing conditions. A survey and some open questions, Probab. Surv., 2, 107-144, (2005), update of, and a supplement to, the 1986 original · Zbl 1189.60077  Bryc, W.; Smoleński, W., Moment conditions for almost sure convergence of weakly correlated random variables, Proc. Amer. Math. Soc., 119, 629-635, (1993) · Zbl 0785.60018  Carlson, F., Contributions à la théorie des séries de Dirichlet. III, Ark. Mat. Astr. Fys., 23A, 19, (1933) · JFM 48.0338.02  Chatterjee, S.; Soundararajan, K., Random multiplicative functions in short intervals, Int. Math. Res. Not. IMRN, 479-492, (2012) · Zbl 1248.11056  Chow, Y. S.; Teicher, H., Probability theory: independence, interchangeability, martingales, Springer Texts Statist., (1997), Springer-Verlag New York · Zbl 0891.60002  Conway, J. B., Functions of one complex variable, Grad. Texts in Math., vol. 11, (1978), Springer-Verlag New York  Erdős, P., Some unsolved problems, Magyar Tud. Akad. Mat. Kutató Int. Közl., 6, 221-254, (1961) · Zbl 0100.02001  Granville, A.; Soundararajan, K., Pretentious multiplicative functions and an inequality for the zeta-function, (Anatomy of Integers, CRM Proc. Lecture Notes, vol. 46, (2008), Amer. Math. Soc. Providence, RI), 191-197 · Zbl 1187.11032  Häggström, O., Finite Markov chains and algorithmic applications, London Math. Soc. Stud. Texts, vol. 52, (2002), Cambridge University Press Cambridge · Zbl 0999.60001  Halász, G., On random multiplicative functions, (Hubert Delange Colloquium, Orsay, 1982, Publ. Math. Orsay, vol. 83, (1983), Univ. Paris XI Orsay), 74-96  Harper, A. J., Bounds on the suprema of Gaussian processes, and omega results for the sum of a random multiplicative function, Ann. Appl. Probab., 23, 584-616, (2013) · Zbl 1268.60075  Harper, A. J., On the limit distributions of some sums of a random multiplicative function, J. Reine Angew. Math., 678, 95-124, (2013) · Zbl 1274.60067  Hough, B., Summation of a random multiplicative function on numbers having few prime factors, Math. Proc. Cambridge Philos. Soc., 150, 193-214, (2011) · Zbl 1231.11117  Koukoulopoulos, D., On multiplicative functions which are small on average, Geom. Funct. Anal., 23, 1569-1630, (2013) · Zbl 1335.11082  Lau, Y.-K.; Tenenbaum, G.; Wu, J., On mean values of random multiplicative functions, Proc. Amer. Math. Soc., 141, 409-420, (2013) · Zbl 1294.11167  Littlewood, J. E., Quelques conséquences de l’hypothèse que la fonction $$\zeta(s)$$ de Riemann n’a pas de zéros dans le demi-plan $$\mathfrak{R}(s) > \frac{1}{2}$$, C. R. Acad. Sci., 154, 263-266, (1912) · JFM 43.0329.02  Meaney, C., Remarks on the Rademacher-menshov theorem, (CMA/AMSI Research Symposium “Asymptotic Geometric Analysis, Harmonic Analysis, and Related Topics”, Proc. Centre Math. Appl. Austral. Nat. Univ., vol. 42, (2007), Austral. Nat. Univ. Canberra), 100-110 · Zbl 1213.42122  Newman, D. J., Simple analytic proof of the prime number theorem, Amer. Math. Monthly, 87, 693-696, (1980) · Zbl 0444.10033  Royden, H. L., Real analysis, (1988), Macmillan Publishing Company New York · Zbl 0704.26006  Shiryaev, A. N., Probability, Grad. Texts in Math., vol. 95, (1996), Springer-Verlag New York, translated from the first (1980) Russian edition by R.P. Boas · Zbl 0909.01009  Tenenbaum, G., Introduction to analytic and probabilistic number theory, Cambridge Stud. Adv. Math., vol. 46, (1995), Cambridge University Press Cambridge, translated from the second French edition (1995), by C.B. Thomas · Zbl 0788.11001  Titchmarsh, E. C., The theory of the Riemann zeta-function, (1986), The Clarendon Press, Oxford University Press New York, edited and with a preface by D.R. Heath-Brown · Zbl 0601.10026  Viana, M., Ergodic theory of interval exchange maps, Rev. Mat. Complut., 19, 7-100, (2006) · Zbl 1112.37003  Wintner, A., Random factorizations and Riemann’s hypothesis, Duke Math. J., 11, 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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.