⇦Chinese Physics Letters, 2019, Vol. 36, No. 7, Article code 070201 A Proof of First Digit Law from Laplace Transform * Mingshu Cong (丛明舒)1, Bo-Qiang Ma (马伯强)1,2** Affiliations 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 Received 22 March 2019, online 20 June 2019 *Supported by the National Natural Science Foundation of China under Grant No 11475006.
**Corresponding author. Email: mabq@pku.edu.cn Citation Text: Cong M S and Ma B Q 2019 Chin. Phys. Lett. 36 070201    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. DOI:10.1088/0256-307X/36/7/070201 PACS:02.30.Uu, 02.50.-r, 02.50.Cw © 2019 Chinese Physics Society Article Text The first digit law, which is also called the significant digit law or Benford's law, was first noticed by Newcomb in 1881,[1] and then re-discovered independently by Benford in 1938.[2] It is an empirical observation that the first digits of natural numbers are preferentially small rather than a uniform distribution as might be expected. More accurately, the probability that a number begins with digit $d$, where $d=1,2,\ldots,9$, can be expressed as $$ P_d=\log\Big(1+\frac{1}{d}\Big),~~d=1, 2,\ldots, 9,~~ \tag {1} $$ as shown in Fig. 1.
cpl-36-7-070201-fig1.png
Fig. 1. Benford's law of the first digit distribution, from which we can see that the probability of finding numbers with leading digit 1 is larger than that with $2,\ldots, 9$, respectively.
Empirically, the populations of countries, the areas of lakes, the lengths of rivers, the Arabic numbers on the front page of a newspaper,[2] physical constants,[3] the stock market indices,[4] file sizes in a personal computer,[5] etc., all conform to the peculiar law well. Benford's law has been verified to hold true for a vast number of examples in various domains, such as economics,[4] social science,[6] environmental science,[7] biology,[8] geology,[9] astronomy,[10] statistical physics,[11,12] nuclear physics,[13–17] particle physics,[18] and some dynamical systems.[19–21] Also, there have been many explorations on applications of the law in various fields, mainly to detect data and judge their reasonableness, such as in distinguishing and ascertaining fraud in taxing and accounting[22–24] and falsified data in scientific experiments.[25] Benford's law has several elegant properties. It is scale-invariant,[26,27] which means that the law does not depend on any particular choice of units. This law is also base-invariant,[28–30] which means that it is independent of the base $b$ with a general form $$ P_d=\log_{b}\Big(1+\frac{1}{d}\Big),~d=1, 2,\ldots, {b-1},~{\rm for}~b \ge 2.~~ \tag {2} $$ The law is also found to be power-invariant,[18] i.e., any power ($\neq 0$) on numbers in the data set does not change the first digit distribution. Although there have been many studies on Benford's law,[31] the underlying reason for the success of this law has remained elusive for more than 100 years. It was unclear whether Benford's Law was due to some unknown mechanism of nature or whether it is merely a logical consequence of the human number system. However, the situation has changed due to the appearance of a general derivation of Benford's law from the application of the Laplace transform,[32] where a strict version of Benford's law is derived as composed of a Benford term and an err term. The Benford term explains the prevalence of Benford's law and the err term leads to derivations from the law with four categories of number sets. It is the purpose of this work to provide a simpler and more elegant version of the derivation of Benford's law compared to Ref.  [32]. Through this derivation, it is easier to understand the rationality of Benford's law. We reveal that the first digit law can be derived as the main term from the Laplace transform. This explains why Benford's law is so successful for many number sets. We perform similar analyses on the regularities of the second digit and $i$th-significant digit distributions, and extend the law to a more general rule for the first several digit distribution. We also estimate the error term and point out conditions for the validity of this law. For simplification, we constrain ourselves to the decimal system first. Let $F(x)$ be an arbitrary normalized probability density function defined on positive real number set $\mathbb{R}^+$. Here we use the capital letter $F$ instead of the lowercase one, as opposed to the convention. Of course, in the real case, the variable $x$ may be negative or bounded, but this is not harmful to our derivation. When $x$ can be negative, we can use the probability density function of its absolute value, keeping results unchanged. In the decimal system, the probability $P_d$ of finding a number whose first digit is $d$ is the sum of the probability that it is contained in the interval $[d \cdot 10^n, (d+1) \cdot 10^n)$ for all integer $n$, therefore $P_d$ can be expressed as $$ P_d=\sum_{n=-\infty}^{\infty} \int_{d \cdot 10^n}^{(d+1) \cdot 10^n} F(x) dx,~~ \tag {3} $$ which can also be rewritten as $$ P_d=\int_{0}^{\infty} F(x)g_d(x) dx,~~ \tag {4} $$ with $g_d(x)$ being a new density function whose meaning will be clear in the following. Here the lowercase letter is used, due to conventions for Laplace transform in the following. Adopting the Heaviside step function $$ \eta(x)= \left\{\begin{matrix} 1, & {\rm if~} x \ge 0, \\ 0, & {\rm if~} x < 0, \end{matrix} \right.,~~ \tag {5} $$ we can write $g_d(x)$ as $$ g_d(x)=\sum_{n=-\infty}^{\infty}[\eta (x-d \cdot 10^n)- \eta (x-(d+1) \cdot 10^n)].~~ \tag {6} $$ Based on the above discussion, we can understand to some extent why numbers prefer smaller first digits. Naively one might think that the nine digits in the decimal system play the same roles, but they define different density $g_d(x)$ as shown above, thus behave differently in the decimal system. For better illustration, we draw the images of $g_1(x)$ and $g_2(x)$ in the interval $[1,30)$, as shown in Fig. 2, from which we notice that the two density functions have totally different shapes. Neither of them can simply be a translation or an expansion of the other. All the above derivations are rigorous. In fact, using Eq. (3) or (4), we can calculate $P_d$ for any given $F(x)$ numerically. Usually, it does not strictly fit in with Eq. (1). In this sense, Benford's law is not a rigorous 'law' with strong predictive power. However, using the technique of Laplace transform, we show in the following that Benford's law is a rather good approximation for those well-behaved probability density functions.
cpl-36-7-070201-fig2.png
Fig. 2. Images of $g_1(x)$ (upper) and $g_2(x)$ (lower), from which we notice that the gap indicated by the red line in $g_2(x)$ is wider than that in $g_1(x)$. This shows that the distribution of $g_1(x)$ is denser than $g_2(x)$ in the whole number range.
We now prove that if a probability density function has an inverse Laplace transform, it satisfies Benford's law well. Recalling the complex inversion formula,[33] if $F(x)$, extended to the complex plane, satisfies: (1) $F(x)$ is analytic on $\mathbb{C}$ except for a finite number of isolated singularities, (2) $F(x)$ is analytic on the half plane $\{x|{\rm Re}z>0\}$, (3) there are positive constants $M$, $R$, and $\beta$ such that $|F(x)|\le M/|x|^{\beta}$ whenever $|z|\geq R$, $F(x)$ will have an inverse Laplace transform. We can say a probability density function to be 'well-behaved' if it satisfies these three conditions and its inverse Laplace transform is smooth enough, i.e., without violent oscillation. Exponential functions, some fractional functions, and a handful of other common functions are all well-behaved. Thus the derivation in the following has wide application. In what follows, we assume that $F(x)$ is well-behaved. Let $f(t)$ be the inverse Laplace transform of $F(x)$, and $G(t)$ be the Laplace transform of $g(x)$, i.e., $$\begin{align} F(x)=\,&\int_{0}^{\infty} f(t)e^{-tx} dt,~~ \tag {7} \end{align} $$ $$\begin{align} G(t)=\,&\int_{0}^{\infty} g(x)e^{-tx} dx.~~ \tag {8} \end{align} $$ Laplace transforms have the following property $$\begin{align} \int_{0}^{\infty} F(x)g(x) dx =\,& \int_{0}^{\infty}dx g(x) \int_{0}^{\infty} f(t)e^{-tx} dt \\ =\,&\int_{0}^{\infty}dt f(t) \int_{0}^{\infty} g(x)e^{-tx} dx \\ =\,& \int_{0}^{\infty} f(t)G(t) dt,~~ \tag {9} \end{align} $$ which means that Laplace transform can act on either the function $f$ or $g$ with the above integral result keeping unchanged. To derive the left-hand side of the above equation, we would like to calculate the right-hand side instead. Because it is comparably convenient to calculate the Laplace transform of $g_d(x)$, $$\begin{alignat}{1} G_d(t)=\,&\int_{0}^{\infty} g_d(x)e^{-tx} dx \\ =\,&\sum_{n=-\infty}^{\infty} \int_{d \cdot 10^n}^{(d+1) \cdot 10^n}e^{-tx} dx \\ =\,&\frac{1}{t}\sum_{n=-\infty}^{\infty}(e^{-td \cdot 10^n}-e^{-t(d+1) \cdot 10^n}),~~ \tag {10} \end{alignat} $$ which can be treated as a function of two variables $d$ and $t$. Although $d$ is defined on the decimal digit set ${1,2,\ldots,9}$, it can be extended to the whole real axis. Therefore, $G_d(t)$ is a continuous function of $d$ as well as $t$. A technique to evaluate $G_d(t)$ is to calculate its partial derivative with respect to $d$ approximately, and then integrate the partial derivative to derive the result $$\begin{align} \frac{\partial G_d(t)}{\partial d} =\,&\sum_{n=-\infty}^{\infty}(-10^n e^{-td \cdot 10^n}+10^n e^{-t(d+1) \cdot 10^n}) \\ \simeq\,& \int_{ -\infty}^{\infty}(- 10^x e^{-td \cdot 10^x}+ 10^x e^{-t(d+1) \cdot 10^x}) dx \\ =\,& \frac{1}{\rm {ln}10}\int_{0}^{\infty}(-e^{-tdy}+e^{-t(d+1)y}) dy \\ =\,& \frac{1}{\rm{ln}10}\Big(-\frac{1}{td}+\frac{1}{t(d+1)}\Big).~~ \tag {11} \end{align} $$ There is one and only one approximation, i.e., we adopt an integration to replace a summation. Because $G_d(t)\to 0$ when $d\to \infty$, Eq. (11) can be integrated to yield $$ G_d(t)\simeq \frac{1}{t} \log_{10}\Big(1+\frac{1}{d}\Big).~~ \tag {12} $$ Then using Eq. (9), we obtain $$\begin{align} P_d =\,& \int_{0}^{\infty} F(x)g_d(x) dx \\ =\,&\int_{0}^{\infty} G_d(t)f(t) dt \\ \simeq & \int_{0}^{\infty} \frac{f(t)}{t} \log_{10}\Big(1+\frac{1}{d}\Big) dt \\ =\,& \log_{10}\Big(1+\frac{1}{d}\Big) \int_{0}^{\infty} \frac{f(t)}{t} dt \\ =\,& \log_{10}\Big(1+\frac{1}{d}\Big),~~ \tag {13} \end{align} $$ where we have used the following normalization condition of $f(t)$, $$\begin{align} 1 =\,& \int_{0}^{\infty} F(x) dx \\ =\,& \int_{0}^{\infty} dx\int_{0}^{\infty} f(t)e^{-tx} dt \\ =\,& \int_{0}^{\infty} dt f(t)\int_{0}^{\infty} e^{-tx} dx \\ =\,& \int_{0}^{\infty} \frac{f(t)}{t} dt.~~ \tag {14} \end{align} $$ Equation (13) is exactly the first digit law for the decimal system. Thus we show that well-behaved functions satisfy Benford's law approximately. A more rigorous derivation without the approximately equal signs in Eqs. (11)-(13) can be found in Ref.  [32]. Compared to Ref.  [32], the method provided above accords with our intuition better. In fact, unnecessarily complicated treatments are introduced to guarantee the strictness of the proof in Ref.  [32]. For example, a logarithmic scale is adopted after Laplace transform, merely to derive Eq. (12) of Ref.  [32], which corresponds to Eq. (13) in this study. Equation (13), though approximately holds, is set up on the original linear scale, thus manifests itself as a property of the direct Laplace transform, instead of the logarithmic Laplace transform which bears less intuitive physical meanings. In this study, no logarithmic transform is required to derive Benford's law. According to derivations so far, we can already explain the rationality of Benford's law through a clear chain of logic, as follows: (1) The integral of the product of $F(x)$ and $g(x)$ equals the integral of the product of the inverse Laplace transform of $F(x)$ and the Laplace transform of $g(x)$, i.e., Eq. (9). (2) The Laplace transform of $g(x)$ approximately equals the Benford term divided by $t$, i.e., Eq. (12). (3) The normalization condition of $F(x)$ guarantees that the integral of the inverse Laplace transform of $F(x)$ divided by $t$ equals $1$, i.e., Eq. (14). (4) Therefore, the integral of the product of $F(x)$ and $g(x)$ approximately equals the Benford term, i.e., Eq. (13). Such a chain of logic is not apparent in Ref.  [32]. The second significant digit law was also given by Newcomb.[1] In the decimal system, it is $$\begin{align} &P({\rm 2nd~digit~being~}d)\\ =\,&\sum_{k=1}^9\log_{10}(1+(10k+d)^{-1}),\\ &~d=0,1,\ldots,9.~~ \tag {15} \end{align} $$ Hill derived a general $i$th-significant digit law:[30] letting $D_i$ ($D_1,D_2,\ldots$) denote the $i$th-significant digit (with base 10) of a number (e.g., $D_1(0.0314)=3$, $D_2(0.0314)=1$, $D_3(0.0314)=4$), then for all positive integers $k$ and all $d_j \in {0,1,\ldots,9} $, $j=1,2,\ldots, k$, one has $$\begin{align} &P(D_1=d_1,\ldots,D_k=d_k)\\ &=\log_{10}[1+(\sum_{i=1}^k d_i \cdot10^{k-i})^{-1}].~~ \tag {16} \end{align} $$ We propose here a general form of digit law, and show that both the second significant digit law and the general $i$th-significant digit law are only corollaries of this general form. We calculate $P_{b,d,l,k}$, which is the probability that the integer composed of the first $k$ digits (base $b$) of an arbitrary number (e.g., for the number $0.0314$ and $k=2$, this integer is 31) is between $d$ and $d+l$ ($b^{k-1} \le d < d+l < b^k$). Correspondingly we introduce the density function $g_{b,d,l,k}(x)$ as $$ g_{b,d,l,k}(x)=\sum_{n=-\infty}^{\infty}[\eta (x-d \cdot b^n)-\eta (x-(d+l) \cdot b^n)],~~ \tag {17} $$ where the right-hand side is independent of $k$ (while $k$ puts restrictions on $d$ and $l$). Thus we can omit the subscript $k$ in the following. A similar technique gives the Laplace transform of $g_{b,d,l}(x)$, $$ G_{b,d,l}(t)\simeq \frac{1}{t} \log_{b}\Big(1+\frac{l}{d}\Big).~~ \tag {18} $$ Thus we arrive at the general significant digit law $$ P_{b,d,l,k}= \log_{b}\Big(1+\frac{l}{d}\Big).~~ \tag {19} $$ We find that Benford's law (2) corresponds to a special case of this general form for $k=1$ and $l=1$, whereas Hill's general $i$th-significant law (Eq. (16)) corresponds to the case for $b=10$, $d=\sum_{i=1}^k d_i \cdot 10^{k-i}$ and $l= 1$. Newcomb's second significant digit law can be considered as a corollary of Hill's law according to the addition principle in probability theory. We now calculate the error brought by our replacement of the summation to the integration in Eq. (11). Since Eq. (4) is always an accurate expression, the total error is $$ {\it \Delta}_{{\rm total}}=\int_{0}^{\infty} F(x)g_{b,d,l}(x) dx-\log_{b}\Big(1+\frac{l}{d}\Big).~~ \tag {20} $$ If we define $$ {\it \Delta}_{b,d,l}(t)=tG_{b,d,l}(t)-\log_{b}\Big(1+\frac{l}{d}\Big),~~ \tag {21} $$ the total error can be written as $$\begin{align} {\it \Delta}_{{\rm total}} =\,& \int_{0}^{\infty}\frac{ f(t)}{t}[t G_{b,d,l}(t) -\log_{b}\Big(1+\frac{l}{d}\Big)]dt \\ =\,& \int_{0}^{\infty}\frac{ f(t)}{t}{\it \Delta}_{b,d,l}(t)dt.~~ \tag {22} \end{align} $$ Checking the definitions of the two terms of ${\it \Delta}_{b,d,l}$, we can find that the variables of both of them can be multiplied by $b$ and the results remain unchanged, i.e., ${\it \Delta}_{b,d,l}$ is scale invariant. Hence $$ {\it \Delta}_{b,d,l}(bt)={\it \Delta}_{b,d,l}(t).~~ \tag {23} $$ For clarity, we define $$\begin{align} t=\,&e^s,~~ \tag {24} \end{align} $$ $$\begin{align} \widetilde{{\it \Delta}}_{b,d,l}(s)=\,&{\it \Delta}_{b,d,l}(e^s),~~ \tag {25} \end{align} $$ $$\begin{align} \widetilde{f}(s)=\,&f(e^s).~~ \tag {26} \end{align} $$ The corresponding normalization condition is $$\begin{align} \int_{-\infty}^{+\infty}\widetilde{f}(s) ds=1,~~ \tag {27} \end{align} $$ and the property Eq. (23) becomes $$ \widetilde{{\it \Delta}}_{b,d,l}(s+\ln {b})=\widetilde{{\it \Delta}}_{b,d,l}(s).~~ \tag {28} $$ Clearly, $\widetilde{{\it \Delta}}_{b,d,l}$ is a function of period $\ln b$. Furthermore, according to the result for exponential distribution in Ref.  [34] (Corollary 2, $\widetilde{f}(s)$ here is exactly $h_1(x)$ in Ref.  [34], $|\widetilde{{\it \Delta}}_{10,d,1}(s)|$ is $|h_1(x)-\log_{10}(1+\frac{1}{d} )|$ in the equation of Corollary 2), a rather good estimation can be made when $b=10$ and $d= l=1$, $$ 0.029 < \max|\widetilde{{\it \Delta}}_{10,1,1}(s) | < 0.03.~~ \tag {29} $$ We notice that the total error can be expressed as $$\begin{align} {\it \Delta}_{{\rm total}}=\int_{-\infty}^{+\infty}\widetilde{{\it \Delta}}_{b,d,l}(s)\widetilde{f}(s)ds,~~ \tag {30} \end{align} $$ where $\widetilde{f}(s)$ is dependent on $F(x)$ ultimately. In most cases, the correlation between $\widetilde{f}(s)$ and $\widetilde{{\it \Delta}}_{b,d,l}(s)$ is small, thus is the total error. Therefore, Benford's law can be a rather good approximation. However, if $\widetilde{f}(s)$ is close to a periodic function with the exact period $\ln b$, or $\widetilde{f}(s)$ changes signs very fast between positive and negative numbers (this may happen when $F(x)$ is artificially chosen, as the case of telephone numbers in a given region), the small $\widetilde{{\it \Delta}}_{b,d,l}(s)$ is counted and accumulated for many times, therefore the correlation becomes larger. Similar problems also exist for some special types of probability density functions, whose inverse Laplace transforms oscillate violently between positive and negative numbers, e.g., uniform distributions or normal distributions with small variances. Number sets drawn from such distributions, e.g., heights or ages of people, though being natural, still violate Benford's law. By arguing this, we point out that although the above derivation seems quite general, it cannot be universally true. More rigorous discussions about the err term with general applications to four types of number sets can be found in Ref.  [32]. A special case is when the integral of $\widetilde f(s)$ is not only convergent to 1, but also absolutely convergent to a positive real number $M$, then $$\begin{align} {\it \Delta}_{{\rm total}} \le\,&\int_{-\infty}^{+\infty}|\widetilde{{\it \Delta}}_{10,1,1}(s)||\widetilde{f}(s)|ds \\ \le\,& \int_{-\infty}^{+\infty}0.03|\widetilde{f}(s)|ds \\ =\,& 0.03M.~~ \tag {31} \end{align} $$ If $f(s)$ is a positive or negative definite function, it is absolutely integrable. Such an $F(x)$ is called the completely monotonic function in mathematics. This means that $M$ is 1, thus ${\it \Delta}_{{\rm total}}$ is not greater than 0.03. Consequently Benford's law is a good estimation. For example, when $F(x)=\frac{2}{\sqrt{x}}e^{-\sqrt{x}}$, $f(t)=\frac{2}{\sqrt{\pi t}}e^{-\frac{1}{4t}} >0$, and when $F(x)=\frac{4}{(x+1)^5}$, $f(x)=\frac{e^{-t}}{6}>0$. We can assert that in these cases, the total errors are less than 0.03. In fact, numerical results are 0.0005 and 0.009. This verifies our estimation. As a rule of thumb, distributions with monotonic decreasing and relatively smooth probability density functions often conform to Benford's law well.[32] Inverse Laplace transforms of such probability density functions generally change signs only for finite times, thus being absolutely convergent. To understand this, one can view inverse Laplace transform as decomposing the original probability density function into a series of exponential functions, among which some are positive and others negative. If a monotonic decreasing probability density function is relatively smooth, i.e., without a sharp change of probability density, it can be approached mainly by positive exponential distributions, therefore its inverse Laplace transform does not oscillate between positive and negative numbers very much. As an application of this rule of thumb, for non-monotonic decreasing distributions, we can transform them into monotonic decreasing distributions, resulting in better performance of Benford's law, e.g., for normal distributions, we can subtract the mean value from the original data set and obtain a monotonic decreasing distribution. The above calculations and derivations tell us that the significant digit behaviors demonstrate that although our nature has no preference to any specific number, it does have discrimination to digits in numbers as a logical consequence of human counting systems. Therefore, our results justify the conventional wisdom that the violation of Benford's law is a sign that a table of numbers is artificial or anomalous. The underlying reason for the uneven distribution of first digits is due to the basic property of digital system, but not some dynamic source behind nature as people suspected. This also explains why we can use Benford's law to distinguish anomalies or unnaturalness in artificial numbers. The mathematical expressions and derivations provided here are simple, elegant, and all with clear intuitive pictures. They are easily comprehensible. Therefore, this version of proof of Benford's law can also serve as an example for the application of the Laplace transform. The first digit law reveals an astonishing regularity in realistic numbers. We provide in this work a proof of this law from the Laplace transform, and point out the condition for the validity of the law. Compared to Ref.  [32], the derivation here is simple and elegant, and it directly reveals the rationality of the first digit law. From our work, the first digit law is due to the basic structure of the number system. Thus the first digit law is a general rule that applies to vast data sets in the natural world as well as in human social activities. It is no longer strange why Benford's law is so successful in various domains. Such a law should be regarded as a basic mathematical knowledge with great potential for vast applications. References Note on the Frequency of Use of the Different Digits in Natural NumbersBenford’s law and physical constants: The distribution of initial digitsOn the Peculiar Distribution of the U.S. Stock Indexes' DigitsHow do numbers begin? (The first digit law)Survival Distributions Satisfying Benford's LawBenford's Law and the screening of analytical data: the case of pollutant concentrations in ambient airThe number of cells in colonies of the cyanobacterium Microcystis aeruginosa satisfies Benford's lawBenford’s Law Applied to Hydrology Data—Results and Relevance to Other Geophysical DataEmpirical mantissa distributions of pulsarsThe significant digit law in statistical physicsFirst-digit law in nonextensive statisticsAn illustration of Benford's first digit law using alpha decay half livesBenford’s law and half-lives of unstable nucleiBenford's Law and β-Decay Half-LivesBenford’s law and cross-sections of A(n, $ \alpha$ )B reactionsBenford's Law in Nuclear Structure PhysicsFIRST DIGIT DISTRIBUTION OF HADRON FULL WIDTHDo dynamical systems follow Benford’s law?One-dimensional dynamical systems and Benford's lawMulti-dimensional dynamical systems and Benford's LawNot the First Digit! Using Benford's Law to Detect Fraudulent Scientif ic DataOn the Distribution of First Significant DigitsScale-Distortion Inequalities for Mantissas of Finite Data SetsThe Significant-Digit PhenomenonBase-Invariance Implies Benford's LawA Statistical Derivation of the Significant-Digit LawFirst digit law from Laplace transformBenford's law for exponential random variables
[1] Newcomb S 1881 Am. J. Math. 4 39
[2]Benford F 1938 Proc. Am. Philos. Soc. 78 551
[3] Burke J and Kincanon E 1991 Am. J. Phys. 59 952
[4] Ley E 1996 Am. Stat. 50 311
[5] Torres J et al 2007 Eur. J. Phys. 28 L17
[6] Leemis L M et al 2000 Am. Stat. 54 236
[7] Brown R J C 2005 Analyst 130 1280
[8] Costas E et al 2008 Aquat. Bot. 89 341
[9] Nigrini M and Miller S J 2007 Math. Geol. 39 469
[10] Shao L and Ma B Q 2010 Astropart. Phys. 33 255
[11] Shao L and Ma B Q 2010 Physica A 389 3109
[12] Shao L and Ma B Q 2010 Phys. Rev. E 82 041110
[13] Buck B et al 1993 Eur. J. Phys. 14 59
[14] Ni D and Ren Z 2008 Eur. Phys. J. A 38 251
[15] Ni D D et al 2009 Commun. Theor. Phys. 51 713
[16] Liu X J et al 2011 Eur. Phys. J. A 47 78
[17] Jiang H et al 2011 Chin. Phys. Lett. 28 032101
[18] Shao L and Ma B Q 2009 Mod. Phys. Lett. A 24 3275
[19] Tolle C R et al 2000 Chaos 10 331
[20] Berger A et al 2005 Trans. Am. Math. Soc. 357 197
[21] Berger A 2005 Discrete Contin. Dyn. Syst. A 13 219
[22]Nigrini M J 1996 J. Am. Tax. Assoc. 18 72
[23]Nigrini M J 1999 Intern. Auditor 56 21
[24]Rose A M and Rose J M 2003 J. Accountancy 196 58
[25] Diekmann A 2007 J. Appl. Stat. 34 321
[26] Pinkham R S 1961 Ann. Math. Stat. 32 1223
[27] Berger A et al 2008 J. Theor. Probab. 21 97
[28] Hill T P 1995 Am. Math. Mon. 102 322
[29] Hill T P 1995 Proc. Am. Math. Soc. 123 887
[30] Hill T P 1995 Stat. Sci. 10 354
[31]Berger A et al 2009 Benford Online Bibliography http://www.benfordonline.net/
[32] Cong M et al 2019 Phys. Lett. A 383 1836
[33]Marsden J and Hoffman M 1999 Basic Complex Anal. 3rd edn (New York: W. H. Freeman) p 471
[34] Engel H A and Leuenberger C 2003 Stat. Probab. Lett. 63 361