A converse to Halasz's theorem
Number Theory
2011-11-10 v1 Probability
Abstract
We show that the distribution of large values of an additive function on the integers, and the distribution of values of the additive function on the primes are related to each other via a Levy Process. As a consequence we obtain a converse to an old theorem of Halasz. Halasz proved that if f is an strongly additive function with f (p) \in {0, 1}, then f is Poisson distributed on the integers. We prove, conversely, that if f is Poisson distributed on the integers then for most primes p, f(p) = o(1) or f(p) = 1 + o(1).
Cite
@article{arxiv.1109.0037,
title = {A converse to Halasz's theorem},
author = {Maksym Radziwill},
journal= {arXiv preprint arXiv:1109.0037},
year = {2011}
}
Comments
14 pages. Part of my B. Sc. thesis: arxiv:0909.5274