A Proof of First Digit Law from Laplace Transform*

Abstract

The first digit law, also known as Benford's law or the significant digit law, is an empirical phenomenon that the leading digit of numbers from real world sources favors small ones in a form log(1 + 1/d), where d = 1, 2, ..., 9. Such a law keeps elusive for over one hundred years because it was obscure whether this law is due to the logical consequence of the number system or some mysterious mechanism of the nature. We provide a simple and elegant proof of this law from the application of the Laplace transform, which is an important tool of mathematical methods in physics. We reveal that the first digit law is originated from the basic property of the number system, thus it should be attributed as a basic mathematical knowledge for wide applications.

A concise proof of Benford’s law

“…Therefore, it is intuitive to guess that such f (x) with positive c(t) converges to Benford's law, which is exactly the conclusion of Refs. [27,28].…”

“…In this article, first, we aim to derive this law for distributions with Riemann integrable probability density functions on the positive real number set, and the basic idea of the proof is inspired by Refs. [27,28]. Our proof is concise, intuitive, and accessible to anyone with basic knowledge of calculus.…”

“…Research conducted by Engel and Leuenberger [26] proved that exponential functions obey this law within an allowed error range. In 2019, a study at Peking University gave a proof of the law when the probability density function has an inverse Laplace transform [27,28], revealing that the first digit law originates from the basic property of the human number system.…”