Study of Benford's Law Open Access

Zhao, Mengqi (2015)

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Having been studied for over a hundred years, Benford's Law is an anomaly of numbers. It was named after Frank Benford and was introduced in the 19th century. In this paper, we illustrate its counter-intuitive nature that numbers are actually not distributed with equal probability in Section 1. The law started with the study of the leading digit frequency, but expanded to the second digit, the first-two digits, the first-three digits, ..., and to any base beyond base 10. Though the real-world applications of the law were limited before the 20th century, it has been used by auditors, accountants, scientists to detect data fraud in recent times. This will be discussed in Section 1 following the background information. In Section 2, by introducing the concept of uniform distribution mod 1, we will define the notion of a Benford sequence. The next step is to establish the mathematical foundations by defining a good sequence and Weyl's Criterion in the mathematical justification subsection. They provide a strategy to prove whether a sequence conforms to Benford's Law or not. Then the strategy was proved accordingly. Finally, with the established strategy, we take a look at how mathematicians have used it to prove some sequences to be Benford.

Table of Contents

1 Introduction 1

1.1 Historical Background 4

1.2 Real-life Applications 7

2 Mathematical Foundations of Benford's Law 11

2.1 Uniform Distribution Modulo1 11

2.2 Definition of "Benford's Law Base B" 14

2.3 Mathematical Justification of Benford's Law 16

3 Established Cases of Benford's Law 25

3.1 n! 25

3.2 Partition Functions 26

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