A structure theorem in probabilistic number theory

TL;DR

This paper establishes a structure theorem linking the distribution of additive functions on integers and primes, showing that if two such functions share similar large-value probabilities, they are essentially the same.

Contribution

It introduces a novel structure theorem that characterizes the relationship between the distributions of additive functions on integers and primes.

Findings

Additive functions with similar large-value probabilities are identical.

The distribution of additive functions on integers is closely related to their distribution on primes.

The theorem clarifies the interdependence between these distributions.

Abstract

We prove that if two additive functions (from a certain class) take large values with roughly the same probability then they must be identical. This is a consequence of a structure theorem making clear the inter-relation between the distribution of an additive function on the integers, and its distribution on the primes.