Probability of digits by dividing random numbers: a psi and zeta functions approach

This paper begins with the statistics of the decimal digits of $n/d$ with n, d positive integers randomly chosen. Starting with a statement by E. Cesàro on probabilistic number theory, we evaluate, through the Euler psi function, an integral appearing there. Furthermore the probabilistic statement itself is proved, using a different approach. The theorem is then generalized to real numbers and to the alpha-th power of the ratio of integers, via an elementary approach involving the psi function and the Hurwitz zeta function.