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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 Cesāro on probabilistic number theory, see Cesāro (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.

@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 = "", url = "", author = "Alessandro Gambini and Giovanni Mingari Scarpello and Daniele Ritelli", }

Reference Type: Journal Article

Subject Area(s): Probability Theory, Statistics