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Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie

IMPACT FACTOR 2018: 1.859

CiteScore 2018: 1.14

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Source Normalized Impact per Paper (SNIP) 2018: 1.411

Mathematical Citation Quotient (MCQ) 2018: 1.55

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Volume 2013, Issue 678


On the limit distributions of some sums of a random multiplicative function

Adam J. Harper
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  • Department of Pure Mathematics and Mathematical Statistics, Wilberforce Road, Cambridge CB3 0WA, England
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Published Online: 2012-03-23 | DOI: https://doi.org/10.1515/crelle.2012.027


We study sums of a random multiplicative function; this is an example, of number-theoretic interest, of sums of products of independent random variables (chaoses). Using martingale methods, we establish a normal approximation for the sum over those n ≦ x with k distinct prime factors, provided that k = o(log log x) as x → ∞. We estimate the fourth moments of these sums, and use a conditioning argument to show that if k is of the order of magnitude of log log x then the analogous normal limit theorem does not hold. The methods extend to treat the sum over those n ≦ x with at most k distinct prime factors, and in particular the sum over all n ≦ x. We also treat a substantially generalised notion of random multiplicative function.

About the article

Received: 2010-12-20

Revised: 2011-09-06

Published Online: 2012-03-23

Published in Print: 2013-05-01

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2013, Issue 678, Pages 95–124, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: https://doi.org/10.1515/crelle.2012.027.

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