A converse to Halasz's theorem

TL;DR

This paper establishes a connection between the distribution of large values of additive functions on integers and primes via Levy processes, providing a converse to Halasz's theorem about Poisson distribution of such functions.

Contribution

It proves that Poisson distribution of an additive function on integers implies specific behavior of the function on most primes, reversing Halasz's original result.

Findings

Poisson distribution on integers implies f(p) = o(1) or 1 + o(1) for most primes p

Distribution of large values relates to Levy processes

Provides a converse to Halasz's theorem

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).