The recursive algorithm to construct Dirichlet function

Abstract

The problem of determining the total number of rational fractions with values are equal to x is considered. The importance of this problem is demonstrated for the procedures processing statistical data, representing the ratio of two discrete variables with a variable denominator. It is found out, that that required number of fractions is equal to the Dirichlet function value at the x point, and the original rule is proposed to construct it, which has a simple geometric interpretation. A proposed implementation of this algorithm shows its computational efficiency, and its importance is noted for problems requiring generation of relatively prime numbers.