Averaging along Uniform Random Integers

Motivated by giving a meaning to “The probability that a random integer has initial digit $d$”, we define a emph{URI-set} as a random set $E$ of natural integers such that each $≥1$ belongs to $E$ with probability $1/n$, independently of other integers. This enables us to introduce two notions of densities on natural numbers: The emph{URI-density}, obtained by averaging along the elements of $E$, and the emph{local URI-density}, which we get by considering the $k$-th element of $E$ and letting $k$ go to $infty$. 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.