Probability of digits by dividing random numbers: A [formula omitted] and [formula omitted] functions approach

Abstract

This paper begins with the statistics of the decimal digits of n/d with (n,d)∈N2 randomly chosen. Starting with a statement by Cesàro on probabilistic number theory, see Cesàro (1885) [3,4], we evaluate, through the Euler ψ 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 pr=120+12{ψ(r10+1110)−ψ(r10+1)}. The theorem is then generalized to real numbers (Theorem 1, holding a proof of both nd results) and to the αth power of the ratio of integers (Theorem 2), via an elementary approach involving the ψ function and the Hurwitz ζ 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.