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Abstract

Benford's law is an empirical "law'' governing the frequency of leading
digits in numerical data sets. Surprisingly, for mathematical sequences the
predictions derived from it can be uncannily accurate. For example, among the
first billion powers of $2$, exactly $301029995$ begin with digit 1, while the
Benford prediction for this count is $10^9\log_{10}2=301029995.66\dots$.
Similar "perfect hits'' can be observed in other instances, such as the digit
$1$ and $2$ counts for the first billion powers of $3$. We prove results that
explain many, but not all, of these surprising accuracies, and we relate the
observed behavior to classical results in Diophantine approximation as well as
recent deep conjectures in this area.