A converse to Halasz's theorem

Details A converse to Halasz's theorem A converse to Halasz's theorem

2011-08-31

arxiv.org/abs/1109.0037v1

Mathematics

Number Theory

14 pages. Part of my B. Sc. thesis: arxiv:0909.5274

Scientific paper

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).

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