Research PaperNo Access

A local Benford Law for a class of arithmetic sequences

Zhaodong Cai

Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104-6395, USA

E-mail Address: [email protected]

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A. J. Hildebrand

Department of Mathematics, University of Illinois, Urbana, IL 61801, USA

E-mail Address: [email protected]

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Junxian Li

Mathematisches Institut, Georg-August Universitaet Goettingen, Bunsenstrasse 3-5, D-37073 Goettingen, Germany

E-mail Address: [email protected]

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https://doi.org/10.1142/S1793042119500325Cited by:0 (Source: Crossref)

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.

This work originated with an undergraduate research project carried out at the Illinois Geometry Lab in 2016. Some of the experimental results in this paper were generated using the Illinois Campus Computing Cluster, a high performance computing platform at the University of Illinois.

Keywords:

AMSC: 11K31, 11K06, 11N05, 11B05

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