A local Benford Law for a class of arithmetic sequences

Abstract

It is well-known that sequences such as the Fibonacci numbers and the factorials satisfy Benford's Law, that is, leading digits in these sequences occur with frequencies given by $P(d)=\log_{10}(1+1/d)$, $d=1,2,\dots,9$. In this paper, we investigate leading digit distributions of arithmetic sequences from a local point of view. We call a sequence locally Benford distributed of order $k$ if, roughly speaking, $k$-tuples of consecutive leading digits behave like $k$ independent Benford-distributed digits. This notion refines that of a Benford distributed sequence, and it provides a way to quantify the extent to which the Benford distribution persists at the local level. Surprisingly, most sequences known to satisfy Benford's Law have rather poor local distribution properties. In our main result we establish, for a large class of arithmetic sequences, a "best-possible" local Benford Law, that is, we determine the maximal value $k$ such that the sequence is locally Benford distributed of order $k$. The result applies, in particular, to sequences of the form $\{a^n\}$, $\{a^{n^d}\}$, and $\{n^β a^{n^α}\}$, as well as the sequence of factorials $\{n!\}$ and similar iterated product sequences.