Abstract

This paper begins with the statistics of the decimal digits of n/d with n, d 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 (Theorem 1) and to the alpha-th power of the ratio of integers (Theorem 2), via an elementary approach involving the psi function and the Hurwitz zeta 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.