Mathematical Notes

, Volume 88, Issue 3–4, pp 449–463

# Benford’s law and distribution functions of sequences in (0, 1)

Article

## Abstract

Applying the theory of distribution functions of sequences xn ∈ [0, 1], n = 1, 2, ..., we find a functional equation for distribution functions of a sequence xn and show that the satisfaction of this functional equation for a sequence xn is equivalent to the fact that the sequence xn to satisfies the strong Benford law. Examples of distribution functions of sequences satisfying the functional equation are given with an application to the strong Benford law in different bases. Several direct consequences from uniform distribution theory are shown for the strong Benford law.

### Key words

distribution function of a sequence Benford’s law density of occurrence of digits

## Preview

### References

1. 1.
S. Newcomb, “Note on the frequency of use of the different digits in natural numbers,” Amer. J. Math. 4, 39–40 (1881).
2. 2.
F. Benford, “The law of anomalous numbers,” Proc. Amer. Phil. Soc. 78, 551–572 (1938).Google Scholar
3. 3.
R. A. Raimi, “The first digit problem,” Amer. Math. Monthly 83(7), 521–538 (1976).
4. 4.
P. Diaconis, “The distribution of leading digits and uniform distribution mod 1,” Ann. Probab. 5(1), 72–81 (1977).
5. 5.
S. Kunoff, “N! has the first digit property,” Fibonacci Quart. 25(4), 365–367 (1987).
6. 6.
P. Schatte, “On mantissa distribution in computing and Benford’s law,” J. Inform. Process. Cybernet. 24(9), 443–455 (1988).
7. 7.
K. Nagasaka, S. Kanemistu, and J.-S. Shiue, “Benford’s law: the logarithmic law of first digit,” in Colloq. Math. Soc. János Bolyai, Vol. 51: Number Theory: Elementary and Analytic, Proc. Conf., Budapest, 1987 (North-Holland, Amsterdam, 1990), Vol. 51, pp. 361–391.Google Scholar
8. 8.
L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences., in Pure Appl. Math. (John Wiley & Sons, New York, 1974).Google Scholar
9. 9.
M. Drmota and R. F. Tichy, Sequences, Discrepancies and Applications, in Lecture Notes in Math. (Springer-Verlag, Berlin, 1997), Vol. 1651.Google Scholar
10. 10.
O. Strauch and Š. Porubský, Distribution of Sequences: ASampler, in Schr. Slowak. Akad. Wiss. (Peter Lang, Frankfurt am Main, 2005), Vol. 1.Google Scholar
11. 11.
A. I. Pavlov, “On distribution of fractional parts and Benford’s law,” Izv. Akad. Nauk SSSR Ser. Mat. 45(4), 760–774 (1981).
12. 12.
P. Kostyrko, M. Mačaj, T. Šalát, and O. Strauch, “On statistical limit points,” Proc. Amer. Math. Soc. 129(9), 2647–2654 (2001).
13. 13.
G. Pólya and G. Szegő, Aufgaben und Lehrsätze aus der Analysis, Vols. 1 and 2, in Grundlehren Math. Wiss. (Springer-Verlag, Berlin, 1964), Vol. 19.Google Scholar
14. 14.
A. R. Giuliano and O. Strauch, “On weighted distribution functions of sequences,” Unif. Distrib. Theory 3(1), 1–18 (2008).