Limit Theorems for Empirical Density of Greatest Common Divisors

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.

18 pages

References in corpus (2)

• On the probability distribution of the gcd and lcm of $r$-tuples of integers
• Asymptotic normality and greatest common divisors

Cited by in corpus (1)

• Large and moderate deviation principles for Engel continued fractions