Gambini, Alessandro ; Mingari Scarpello, Giovanni ; Ritelli, Daniele
(2012)
*Probability of digits by dividing random numbers: a psi and zeta functions approach.*
Expositiones Mathematicae, 30
(3).
p. 223238.
ISSN 0723-0869

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## 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.

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.

Document type

Article

Creators

Keywords

Elementary probability, Euler psi function, Hurwitz zeta function

Subjects

ISSN

0723-0869

DOI

Deposit date

31 Oct 2012 09:57

Last modified

29 Jan 2013 10:16

URI

## Other metadata

Document type

Article

Creators

Keywords

Elementary probability, Euler psi function, Hurwitz zeta function

Subjects

ISSN

0723-0869

DOI

Deposit date

31 Oct 2012 09:57

Last modified

29 Jan 2013 10:16

URI

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