Averaging along Uniform Random Integers

TL;DR

This paper introduces the URI-set, a probabilistic model for natural numbers, and explores its properties, including Benford's law adherence, independence of mantissae in different bases, and relationships with existing densities.

Contribution

It defines URI-density and local URI-density, proving their properties, and establishes their connections with known densities and Benford's law.

Findings

URI-set elements follow Benford's law in both densities.

Mantissae in multiplicatively independent bases are independent.

URI-density is equivalent to log-density and stronger than natural density.

Abstract

Motivated by giving a meaning to "The probability that a random integer has initial digit d", we define a URI-set as a random set E of natural integers such that each n>0 belongs to E with probability 1/n, independently of other integers. This enables us to introduce two notions of densities on natural numbers: The URI-density, obtained by averaging along the elements of E, and the local URI-density, which we get by considering the k-th element of E and letting k go to infinity. We prove that the elements of E satisfy Benford's law, both in the sense of URI-density and in the sense of local URI-density. Moreover, if b_1 and b_2 are two multiplicatively independent integers, then the mantissae of a natural number in base b_1 and in base b_2 are independent. Connections of URI-density and local URI-density with other well-known notions of densities are established: Both are stronger than the natural density, and URI-density is equivalent to log-density. We also give a stochastic interpretation, in terms of URI-set, of the H_\infty-density.