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 the origin of Benford's law is from the way that we write numbers, thus should be taken as a basic mathematical knowledge.

Highlights

  • There is an empirical law concerning the occurrence of the first digits in real-world data, stating that the first digits of natural numbers prefer small ones rather than a uniform distribution as might be expected

  • The first digit law is applied in computer science for speeding up calculation [29], minimizing expected storage space [30, 31], analyzing the behavior of floating-point arithmetic algorithms [31], and for Preprint submitted to PLA, and published as: M

  • The first digit law has revealed an astonishing regularity of natural number sets

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Summary

Introduction

There is an empirical law concerning the occurrence of the first digits in real-world data, stating that the first digits of natural numbers prefer small ones rather than a uniform distribution as might be expected. We can safely assert that the deviation from Benford’s law is always less than a small proportion of the L1-norm of the logarithmic inverse Laplace transform of the probability density function. We introduce a guideline to judge how well a specific distribution obeys Benford’s law In this method, the degree of deviation from the law is associated with the oscillatory behavior of the probability density function in the inverse Laplace space.

The intuition
The Laplace transform of the digital indicator function
The derivation of the general digit law
The error term
Small- f 1 distribution
Scale-invariant distribution
Completely monotonic distribution
Summary
Full Text
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