Chin. Phys. Lett.  2019, Vol. 36 Issue (7): 070201    DOI: 10.1088/0256-307X/36/7/070201
A Proof of First Digit Law from Laplace Transform
Mingshu Cong1, Bo-Qiang Ma1,2**
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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Mingshu Cong
Bo-Qiang Ma
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