Uniform Distribution Theory 7(2), pp. 35–60.

**ISSN/ISBN:** Not available at this time.
**DOI:** Not available at this time.

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**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.

**Bibtex:**

```
@article{,
title={Averaging along uniform random integers},
author={Janvresse, {\'E}lise and De La Rue, Thierry},
journal={Uniform Distribution Theory},
volume={7},
issue={2},
year={2012},
pages={35--60},
}
```

**Reference Type:** Journal Article

**Subject Area(s):** Analysis