### Gambini, A, Scarpello, GM and Ritelli, D (2012)

#### Probability of digits by dividing random numbers: A ψ and ζ functions approach

Expositiones Mathematicae, Vol. 30, No. 3, pp. 223–238.

**ISSN/ISBN:** Not available at this time.
**DOI:** 10.1016/j.exmath.2012.03.001

**Abstract:** This paper begins with the statistics of the decimal digits of n/d with (n,d)in N^2 randomly chosen. Starting with a statement by Cesàro on probabilistic number theory, see Cesàro (1885) [3] and [4], we evaluate, through the Euler psi function, an integral appearing there. Furthermore the probabilistic statement itself is proved, using a different approach: in any case the probability of a given digit r to be the first decimal digit after dividing a couple of random integers is
p_r=1/20+1/2{psi(r/10+11/10)-psi(r/10+1)}.
The theorem is then generalized to real numbers (Theorem 1, holding a proof of both nd results) and to the ?th power of the ratio of integers (Theorem 2), via an elementary approach involving the ? function and the Hurwitz ? function. The article provides historic remarks, numerical examples, and original theoretical contributions: also it complements the recent renewed interest in Benford’s law among number theorists.

**Bibtex:**

```
@article{,
title = "Probability of digits by dividing random numbers: A psi and zeta functions approach ",
journal = "Expositiones Mathematicae ",
volume = "30",
number = "3",
pages = "223 - 238",
year = "2012",
note = "",
issn = "0723-0869",
doi = "http://dx.doi.org/10.1016/j.exmath.2012.03.001",
url = "http://www.sciencedirect.com/science/article/pii/S0723086912000230",
author = "Alessandro Gambini and Giovanni Mingari Scarpello and Daniele Ritelli",
}
```

**Reference Type:** Journal Article

**Subject Area(s):** Probability Theory, Statistics