Regular sets and conditional density: an extension of Benford’s law. (English) Zbl 1092.11009

It is known that the set of prime numbers obeys Benford’s law. It is shown that Benford’s law holds for some subsets of positive integers other than the set of prime numbers by S. Kanemitsu, K. Nagasaka, G. Rauzy and J.-S. Shiue [Lect. Notes Math. 1299, 158–169 (1988; Zbl 0642.10007)].
In this paper the authors propose an extension of Benford’s law, by using the notation of conditional density introduced by A. Fuchs and G. Letta [J. Comb. 3, 601–607 (1996; Zbl 0853.11006)]. It is shown that for a large class of subsets of positive integers, the upper and lower arithmetic and logarithmic densities coincide with the corresponding conditional densities with respect to the set of prime numbers. In this work, the set of prime numbers can be replaced by any regular subset of positive integers.


11B05 Density, gaps, topology
11K99 Probabilistic theory: distribution modulo \(1\); metric theory of algorithms
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