A concise proof of Benford's law

TL;DR

This paper offers an accessible, elegant proof of Benford's law based on Riemann integrable densities, and introduces a criterion to determine if a distribution follows the law, aiding fraud detection.

Contribution

It provides a concise, intuitive proof of Benford's law and a practical criterion for identifying distributions that obey it.

Findings

Proof applies to distributions with Riemann integrable densities

Criterion simplifies checking if a distribution follows Benford's law

Highlights the law's origin from human number system properties

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 the human number system. The criterion can bring great convenience to the field of fraud detection.