On the limit distributions of some sums of a random multiplicative function
-
Adam J. Harper
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.
©[2013] by Walter de Gruyter Berlin Boston
Articles in the same Issue
- Deformation of the O'Grady moduli spaces
- Maximal and reduced Roe algebras of coarsely embeddable spaces
- Einstein manifolds and extremal Kähler metrics
- On the limit distributions of some sums of a random multiplicative function
- The index of a transverse Dirac-type operator: the case of abelian Molino sheaf
- Traces on symmetrically normed operator ideals
- On commutative, operator amenable subalgebras of finite von Neumann algebras
- Convergence of the Kähler–Ricci flow on Fano manifolds
Articles in the same Issue
- Deformation of the O'Grady moduli spaces
- Maximal and reduced Roe algebras of coarsely embeddable spaces
- Einstein manifolds and extremal Kähler metrics
- On the limit distributions of some sums of a random multiplicative function
- The index of a transverse Dirac-type operator: the case of abelian Molino sheaf
- Traces on symmetrically normed operator ideals
- On commutative, operator amenable subalgebras of finite von Neumann algebras
- Convergence of the Kähler–Ricci flow on Fano manifolds
