Limit Theorems for Empirical Density of Greatest Common Divisors

TL;DR

This paper investigates the large deviations and convergence rates of the empirical density of pairs of integers with a given GCD, extending classical number theory results to probabilistic large deviation principles.

Contribution

It introduces large deviation analysis and convergence rate results for the empirical GCD density, expanding the understanding of its probabilistic behavior.

Findings

Established large deviation principles for empirical GCD density

Derived convergence rates to the normal distribution in the CLT

Provided generalizations of classical number theory results

Abstract

The law of large numbers for the empirical density for the pairs of uniformly distributed integers with a given greatest common divisor is a classic result in number theory. In this paper, we study the large deviations of the empirical density. We will also obtain a rate of convergence to the normal distribution for the central limit theorem. Some generalizations are provided.