Probability of digits by dividing random numbers: A ψψ and ζζ functions approach

This paper begins with the statistics of the decimal digits of n/dn/d with (n,d)∈N2(n,d)∈N2 randomly chosen. Starting with a statement by Cesàro on probabilistic number theory, see Cesàro (1885) [3] and [4], we evaluate, through the Euler ψψ 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 rr 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 ααth power of the ratio of integers (Theorem 2), via an elementary approach involving the ψψ function and the Hurwitz ζζ 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.