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

  • Adam J. Harper EMAIL logo

Abstract

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.

Received: 2010-12-20
Revised: 2011-09-06
Published Online: 2012-03-23
Published in Print: 2013-05

©[2013] by Walter de Gruyter Berlin Boston

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