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.
Probability of digits by dividing random numbers: a psi and zeta functions approach / Gambini A.; Mingari Scarpello G.; Ritelli D.. - In: EXPOSITIONES MATHEMATICAE. - ISSN 0723-0869. - STAMPA. - 30:(2012), pp. 223-238. [10.1016/j.exmath.2012.03.001]
Probability of digits by dividing random numbers: a psi and zeta functions approach
GAMBINI, ALESSANDRO;MINGARI SCARPELLO, GIOVANNI;RITELLI, DANIELE
2012
Abstract
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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.