# First digit law from Laplace transform

## TL;DR

This paper explains the ubiquity of Benford's law by analyzing the Laplace transform of digital indicator functions, revealing that the law stems from fundamental properties of number representation.

## Contribution

It introduces a Laplace transform-based framework to understand Benford's law, linking it to the logarithmic Laplace spectrum and monotonic distributions.

## Key findings

- The Laplace spectrum of the digital indicator function is approximately constant.
- Deviations from Benford's law occur with oscillating distributions in inverse Laplace space.
- All completely monotonic distributions approximately satisfy Benford's law.

## Abstract

The occurrence of digits 1 through 9 as the leftmost nonzero digit of numbers from real-world sources is distributed unevenly according to an empirical law, known as Benford's law or the first digit law. It remains obscure why a variety of data sets generated from quite different dynamics obey this particular law. We perform a study of Benford's law from the application of the Laplace transform, and find that the logarithmic Laplace spectrum of the digital indicator function can be approximately taken as a constant. This particular constant, being exactly the Benford term, explains the prevalence of Benford's law. The slight variation from the Benford term leads to deviations from Benford's law for distributions which oscillate violently in the inverse Laplace space. We prove that the whole family of completely monotonic distributions can satisfy Benford's law within a small bound. Our study suggests that Benford's law originates from the way that we write numbers, thus should be taken as a basic mathematical knowledge.