Published in Probability Surveys, Volume 8, January 1, 2011, pages 1-126.
Copyright © 2011 by Arno Berger and Theodore P. Hill. Published in Probability Surveys.
NOTE: At the time of publication, the author Theodore P. Hill was not yet affiliated with Cal Poly.
This work is licensed under a Creative Commons Attribution 3.0 Unported License.
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The definitive version is available at https://doi.org/10.1214/11-PS175.
Drawing from a large, diverse body of work, this survey presents a comprehensive and unified introduction to the mathematics underlying the prevalent logarithmic distribution of significant digits and significands, often referred to as Benford’s Law (BL) or, in a special case, as the First Digit Law. The invariance properties that characterize BL are developed in detail. Special attention is given to the emergence of BL in a wide variety of deterministic and random processes. Though mainly expository in nature, the article also provides strengthened versions of, and simplified proofs for, many key results in the literature. Numerous intriguing problems for future research arise naturally.