Probability Surveys

A basic theory of Benford’s Law

Arno Berger and Theodore P. Hill

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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.

Article information

Probab. Surveys Volume 8 (2011), 1-126.

First available in Project Euclid: 28 July 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60-01: Instructional exposition (textbooks, tutorial papers, etc.) 11K06: General theory of distribution modulo 1 [See also 11J71] 37M10: Time series analysis 39A60: Applications
Secondary: 37A45: Relations with number theory and harmonic analysis [See also 11Kxx] 60F15: Strong theorems 62E10: Characterization and structure theory

Benford’s Law significant digits uniform distribution mod 1 scale-invariance base-invariance sum-invariance shadowing difference equation random probability measure mixture of distributions


Berger, Arno; Hill, Theodore P. A basic theory of Benford’s Law. Probab. Surveys 8 (2011), 1--126. doi:10.1214/11-PS175.

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