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HELSON’S PROBLEM FOR SUMS OF A RANDOM MULTIPLICATIVE FUNCTION

Part of: Multiplicative number theory Stochastic processes Harmonic analysis in several variables Holomorphic functions of several complex variables

Published online by Cambridge University Press:  21 October 2015

Andriy Bondarenko  and
Kristian Seip
Andriy Bondarenko
Affiliation:
Department of Mathematical Analysis, Taras Shevchenko National University of Kyiv, Volodymyrska 64, 01033 Kyiv, Ukraine Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway email andriybond@gmail.com
Kristian Seip
Affiliation:
Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway email seip@math.ntnu.no

Abstract

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We consider the random functions $S_{N}(z):=\sum _{n=1}^{N}z(n)$, where $z(n)$ is the completely multiplicative random function generated by independent Steinhaus variables $z(p)$. It is shown that $\mathbb{E}|S_{N}|\gg \sqrt{N}(\log N)^{-0.05616}$ and that $(\mathbb{E}|S_{N}|^{q})^{1/q}\gg _{q}\sqrt{N}(\log N)^{-0.07672}$ for all $q>0$.

MSC classification

Primary: 11N60: Distribution functions associated with additive and positive multiplicative functions
Secondary: 32A70: Functional analysis techniques 42B30: $H^p$-spaces 60G50: Sums of independent random variables; random walks

Information

Type
Research Article
Information
Mathematika , Volume 62 , Issue 1 , 2016 , pp. 101 - 110
Copyright
Copyright © University College London 2015 

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