**Benford's law** [1] crops up in the
most unexpected places. If you are not familiar with it, an
informal place to get started may be Netflix's documentary
"Digits" from the series
"Connected:
The Hidden Science of Everything".

Why does Benford's law
happen at all? Is there a more general law behind it? Using
**scale-invariance** considerations, Pietronero et al. first
discovered that a more general law is
possible [2], and Barabesi and Pratelli then
extended their work [3]. In our
paper [4] we have in turn rediscovered and
further extended these generalised results, and even uncovered a
previously unknown result for the classic Benford distribution: a
closed-form expression for the distribution of the j-th most
significant digit (for j>=2).

The quintessential application of Benford's law is **forensic
analysis**. In [4], we also question the
optimality of using most significant digits in forensic
detection tests: we show that every time that Benford's "law"
holds for a dataset, an alternative "law" for the **continued
fraction coefficients** of the logarithm of the same data also
holds. That is to say, there is an analogous of Benford's law
for continued fraction coefficients.

In [5] we provide general models for most significant digits and leading continued fraction coefficients. The latter can provide alternative forensic tests, equivalent to using most significant digits.

See the presentation that we gave in EUSIPCO 2021. You can also download a Matlab toolbox reproducing all the results in [5] and [4]. Enjoy!

Félix Balado and Guénolé Silvestre

[1] F. Benford, "The law of anomalous numbers", Proceedings of the American Philosophical Society, 78(4):551--572, 1938.

[2] L. Pietronero, E. Tosatti, V. Tosatti, and A. Vespignani, "Explaining the uneven distribution of numbers in nature: the laws of Benford and Zipf", Physica A: Statistical Mechanics and its Applications, 293(1):297--304, 2001.

[3] L. Barabesi and L. Pratelli, "On the generalized Benford law", Statistics & Probability Letters, 160:108702, 2020.

[4] F. Balado and G. Silvestre, "Benford's Law: Hammering a Square Peg Into a Round Hole?", 29th European Conference on Signal Processing (EUSIPCO), Dublin, Ireland, August, 2021, pp. 796--800. [full text]

[5] F. Balado and G. Silvestre,
"General Distributions of Number Representation
Elements", arXiv:2301.10547
To appear in *Probability in the Engineering and Informational
Sciences* [full text] (Cambridge University Press), 2024.

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