Skip to main page content
U.S. flag

An official website of the United States government

Dot gov

The .gov means it’s official.
Federal government websites often end in .gov or .mil. Before sharing sensitive information, make sure you’re on a federal government site.

Https

The site is secure.
The https:// ensures that you are connecting to the official website and that any information you provide is encrypted and transmitted securely.

Access keys NCBI Homepage MyNCBI Homepage Main Content Main Navigation
. 2023 Jan 13;4(4):841-844.
doi: 10.1016/j.fmre.2023.01.002. eCollection 2024 Jul.

A concise proof of Benford's law

Luohan Wang  1 , Bo-Qiang Ma  1   2   3
Affiliations

A concise proof of Benford's law

Luohan Wang et al. Fundam Res. .

Abstract

This article presents a concise proof of the famous Benford's law when the distribution has a Riemann integrable probability density function and provides a criterion to judge whether a distribution obeys the law. The proof is intuitive and elegant, accessible to anyone with basic knowledge of calculus, revealing that the law originates from the basic property of human number system. The criterion can bring great convenience to the field of fraud detection.

Keywords: Benford’s law; Criterion; First-digit law; Proof; Significant digit law.

PubMed Disclaimer

Conflict of interest statement

The authors declare that they have no conflicts of interest in this work.

Figures

None
Graphical abstract
Fig. 1
Fig. 1
Graph ofP(d)and the range of error. The red part is the maximum |Er(f)| of f(x) with different λ for each d, and the black dots are Benford’s distributions.
See this image and copyright information in PMC

References

    1. Newcomb S. Note on the frequency of use of the different digits in natural numbers. Am. J. Math. 1881;4(1–4):39–40.
    1. Benford F. The law of anomalous numbers. Proc. Am. Philos. Soc. 1938;78(4):551–572.
    1. Berger A., Hill T.P. Princeton University Press; 2015. An Introduction to Benford’s law.
    1. Diaconis P. The distribution of leading digits and uniform distribution mod 1. Ann. Probab. 1977;5(1):72–81.
    1. Luque B., Lacasa L. The first-digit frequencies of prime numbers and Riemann zeta zeros. Proc. R. Soc. A. 2009;465(2107):2197–2216.

LinkOut - more resources