A Proof of First Digit Law from Laplace Transform

doi:10.1088/0256-307X/36/7/070201

Chin. Phys. Lett.

2019, Vol. 36

Issue (7): 070201 DOI: 10.1088/0256-307X/36/7/070201

GENERAL

A Proof of First Digit Law from Laplace Transform

Mingshu Cong¹, Bo-Qiang Ma^1,2**

¹School of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871
²Center for High Energy Physics, Peking University, Beijing 100871

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Mingshu Cong, Bo-Qiang Ma 2019 Chin. Phys. Lett. 36 070201

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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 $d=1, 2,\ldots, 9$. Such a law has been elusive for over 100 years because it has been obscure whether this law is due to the logical consequence of the number system or some mysterious mechanism of 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. It is revealed that the first digit law originates from the basic property of the number system, thus it should be attributed as a basic mathematical knowledge for wide applications.

Received: 22 March 2019 Published: 20 June 2019

PACS:	02.30.Uu	(Integral transforms)
	02.50.-r	(Probability theory, stochastic processes, and statistics)
	02.50.Cw	(Probability theory)

Fund: Supported by the National Natural Science Foundation of China under Grant No 11475006.

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https://cpl.iphy.ac.cn/10.1088/0256-307X/36/7/070201 OR https://cpl.iphy.ac.cn/Y2019/V36/I7/070201

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	Mingshu Cong
	Bo-Qiang Ma

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