Averaging Along Uniform Random Integers

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 ≥ 1 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 ∞. 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 b1 and b2 are two multiplicatively independent integers, then the mantissae of a natural number in base b1 and in base b2 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∞-density.