The distribution of arithmetic functions of general random integers

TL;DR

This paper investigates the limiting distributions of various arithmetic functions applied to a broad class of random integers, including those arising in number theory and geometry.

Contribution

It introduces a general framework for analyzing the distribution of multiplicative and additive functions on diverse classes of random integers.

Findings

Many arithmetic functions of these random integers have well-defined limiting distributions.

The framework applies to integers like Mersenne numbers, Ramanujan tau-function, and curvatures in circle packings.

Results unify understanding of distributional properties across different types of random integers.

Abstract

We consider very general "random integers" and (attempt to) prove that many multiplicative and additive functions of such integers have limiting distributions. These integers include, for instance, the curvatures of Apollonian circle packings, trace of Frobenius elements for elliptic curves, the Ramanujan tau-function, Mersenne numbers, and many others.