Narrow Results

This paper develops a new description of the asymptotics for the empirical distributions of significands and significant digits associated with $(p_n)$, where $p_n$ denotes the $n$th prime number. The work utilizes the space of probability measures on the significand, endowed with a suitable Kantorovich metric, as well as finite-dimensional projections thereof. For sequences sufficiently close to $(p_n)$, it is shown that the limit points of the associated empirical distributions form a circle that is made up of all rescalings of a single absolutely continuous distribution, and is centered at a distribution known as Benford’s law (BL). The precise rate of convergence to that circle is determined. Moreover, even in the infinite-dimensional setting of significands the convergence is seen to occur along a distinguished low-dimensional object, in fact, along a smooth curve intimately related to BL. By connecting $(p_n)$ and BL in a new way, the results rigorously confirm well-documented experimental observations and complement known facts in the literature.

A phenomenological law, called Benford's law, states that the occurrence of the first digit, i.e. 1, 2,…, 9, of numbers from many real world sources is not uniformly distributed, but instead favors smaller ones according to a logarithmic distribution. We investigate, for the first time, the first digit distribution of the full widths of mesons and baryons in the well-defined science domain of particle physics systematically, and find that they agree excellently with the Benford distribution. We also discuss several general properties of Benford's law, i.e. the law is scale-invariant, base-invariant and power-invariant. This means that the lifetimes of hadrons also follow Benford's law.

Benford’s Law is an interesting and unexpected empirical phenomenon — that if we take a large list of number from real data, the first digits of these numbers follow a certain non-uniform distribution. This law is actively used in economics and finance to check that the data in financial reports are real — and not improperly modified by the reporting company. The first challenge is that the cheaters know about it, and make sure that their modified data satisfies Benford’s law. The second challenge related to this law is that lately, another application of this law has been discovered — namely, an application to deep learning, one of the most effective and most promising machine learning techniques. It turned out that the neurons’ weights obey this law only at the difficult-to-detect stage when the fitting is optimal – and when further attempts attempt to fit will lead to the undesirable over-fitting. In this paper, we provide a possible solution to both challenges: we show how to use this law to make financial cheating practically impossible, and we provide qualitative explanation for the effectiveness of Benford’s Law in machine learning.

Benford’s Law, a rule concerning first digits of an array of numbers, has frequently been used to test for the reporting quality of financial statements. When applied to the recent experience for Russian banks, one conclusion is that the 2004 regime shift in accounting standards produced higher quality financial statements. Prior to 2004, the Benford evidence suggests that Russian banks tended to round revenues up, expenses down and thus overstate net income. It also appears that banks may have presented stronger balance sheets than warranted. In the second part of the analysis, the practical use of Benford’s Law to discern a looming bank failure appears limited. While there is, perhaps, some beneficial information to be drawn from testing for Benford distribution conformity, in isolation the tests for financial statement manipulation are inconclusive. Instead, Benford might be used with other early warning detection algorithms to recognize impending bank failures.

We investigate how Korean officers manage revenue and earnings to achieve cognitive reference points (round up). In Korea, revenue has traditionally served as one of the key financial statement figures. Thus, we study revenue management around cognitive reference points both in isolation and to influence earnings around cognitive reference points through a chain effect. Our study compares the distributions of the second (from the left) and first digits in revenue and various proxies for earnings with their corresponding Benford distributions (Benford, 1938). Also, we perform a logistic regression analysis and compute probabilities based on this analysis. The results show that revenue observations have more first digit 1’s and second digit 0’s and fewer first and second digits 9’s than under a Benford distribution. Korean managers appear to round up revenues with high second digits to improve first digits. In addition, we document that revenue observations with second digits of 0 are associated with higher proportions of positive earnings (gross margin and earnings from operations) with second digits of 0. This suggests that Korean firms simultaneously convert the second digits of revenue and earnings to improve the first digits of those numbers. Results from additional tests convey more upward management of the second digit of revenue for firms that have characteristics that indicate higher ex ante benefits and stronger ex post effects from revenue management.

A generalized shadowing lemma is used to study the generation of Benford sequences under non-autonomous iteration of power-like maps T_j : x ↦ α_jx^β_j (1 - f_j(x)), with α_j, β_j > 0 and f_j ∈ C¹, f_j(0) = 0, near the fixed point at x = 0. Under mild regularity conditions almost all orbits close to the fixed point asymptotically exhibit Benford's logarithmic mantissa distribution with respect to all bases, provided that the family (T_j) is contracting on average, i.e. . The technique presented here also applies if the maps are chosen at random, in which case the contraction condition reads 𝔼 log β > 0. These results complement, unify and widely extend previous work. Also, they supplement recent empirical observations in experiments with and simulations of deterministic as well as stochastic dynamical systems.

The United States (U.S.) environmental regulatory system relies heavily on self-reports to assess compliance among regulated facilities. However, the regulatory agencies have expressed concerns regarding the potential for fraud in self-reports and suggested that the likelihood of detection in the federal and state enforcement processes is low. In this paper, we apply Benford’s Law to three years of self-reported discharge parameters from wastewater treatment plant facilities in one U.S. state. We conclude that Benford’s Law alone may not be a reliable method for detecting potential data mishandling for individual facility–parameter combinations, but may provide information about the types of parameters most likely to be fraudulently reported and types of facilities most likely to do so. From a regulatory perspective, this information may help to prioritise potential fraud risks in self reporting and better direct limited resources.

We consider a large class of fast growing sequences of numbers U_n like the nth superfactorial , the nth hyperfactorial and similar ones. We show that their mantissas are distributed following Benford's law in the sense of the natural density. We prove that this is also verified by , by and is passed down to all the sequences obtained by iterating this design process. We also consider the superprimorial numbers and the products of logarithms of integers.

Let $f(z)={\sum }_{n=1}^{\infty }{\lambda }_f(n)e^{2\pi inz}\in S_k^{\mathrm{\text{new}}}({\mathrm{\Gamma }}_0(N))$ be a newform of even weight $k\ge 2$ on ${\mathrm{\Gamma }}_0(N)$ without complex multiplication. Let $\mathbb{P}$ denote the set of all primes. We prove that the sequence ${\{{\lambda }_f(p)\}}_{p\in \mathbb{P}}$ does not satisfy Benford’s Law in any integer base $b\ge 2$. However, given a base $b\ge 2$ and a string of digits $S$ in base $b$, the set

It is well known that sequences such as the Fibonacci numbers and the factorials satisfy Benford’s Law; that is, leading digits in these sequences occur with frequencies given by $P(d)={log}_{10}(1+1/d)$, $d=1,2,\dots ,9$. In this paper, we investigate leading digit distributions of arithmetic sequences from a local point of view. We call a sequence locally Benford distributed of order $k$ if, roughly speaking, $k$-tuples of consecutive leading digits behave like $k$ independent Benford-distributed digits. This notion refines that of a Benford distributed sequence, and it provides a way to quantify the extent to which the Benford distribution persists at the local level. Surprisingly, most sequences known to satisfy Benford’s Law have rather poor local distribution properties. In our main result we establish, for a large class of arithmetic sequences, a “best-possible” local Benford Law; that is, we determine the maximal value $k$ such that the sequence is locally Benford distributed of order $k$. The result applies, in particular, to sequences of the form $\{a^n\}$, $\{a^{n^d}\}$, and $\{n^{\beta }{a}^{n^{\alpha }}\}$, as well as the sequence of factorials $\{n!\}$ and similar iterated product sequences.

Previous studies show that because of cognitive reference points, managers tend to conduct a particular kind of earnings management to affect the cognition of users of financial information. In order to examine whether this kind of earnings management exists in Chinese companies, this paper used Benford's law, which is used increasingly frequently in financial statistics, and conducted a survey on the related financial data of 1,443 listed companies in Chinese A-share market, and studied the current situation of earnings management in Chinese listed companies. The results show that this earnings management based on cognitive reference points also exists in Chinese companies. This paper explains the motive of earnings management from the perspective of cognitive psychology and provides a new method for the recognition and measurement of earnings management.