Hostname: page-component-76fb5796d-25wd4 Total loading time: 0 Render date: 2024-04-26T07:37:30.894Z Has data issue: false hasContentIssue false
  • English
  • Français

Summation of a random multiplicative function on numbers having few prime factors

Published online by Cambridge University Press:  13 December 2010

BOB HOUGH
BOB HOUGH*
Affiliation:
Department of Mathematics, Stanford University, Building 380, Stanford, CA 94305, U.S.A. e-mail: rdhough@stanford.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Given a ±1 random completely multiplicative function f, we prove by estimating moments that the limiting distribution of the normalized sum converges to the standard Gaussian distribution as x → ∞ when r restricts summation to n having o(log log log x) prime factors. We also give an upper bound for the large deviations of with the sum restricted to numbers having a fixed number k of prime factors.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 150 , Issue 2 , March 2011 , pp. 193 - 214
Copyright
Copyright © Cambridge Philosophical Society 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Feller, W.An Introduction to Probability and its Applications, volume 1 (John Wiley and Sons, 1950).Google Scholar
[2]Granville, A. and Soundararajan, K.Large character sums. J. Amer Math Soc., 14 (2) (2001), 365–397.CrossRefGoogle Scholar
[3]Halasz, G. On random multipicative functions. In Hubert Delange Colloquium (Orsay, 1982), volume 83, pages 74–96, Orsay, 1983 (Univ. Paris XI, Publ. Math. Orsay).Google Scholar
[4]Harper, A. On the limit distributions of some sums of a random multiplicative function. Private communication (2009).Google Scholar
[5]Landau, E.Handbuch der Lehre von der Verteilung der Primazahlen (Teubner, 1909).Google Scholar
[6]Tenenbaum, G.Introduction to Analytic and Probabilistic Number Theory (Cambridge University Press, 1995).Google Scholar