Smarandache Type Function Obtained by Duality

TL;DR

This paper extends the Smarandache function to rational numbers, explores its properties as a generating function, and investigates its connections with Euler's totient and Riemann zeta functions.

Contribution

It introduces a novel extension of the Smarandache function to rationals and analyzes its relationships with key number-theoretic functions.

Findings

Extended Smarandache function to rational numbers

Established connections with Euler totient and Riemann zeta functions

Provided a procedure to construct related functions

Abstract

In this paper we extend the Smarandache function from the set $N*$ of positive integers to the set $Q$ pf rational numbers. Using the inverse formula, this function is also regarded as a generating function. We put in evidence a procedure to construct a (numerical) function starting from a given function in two particular cases. Also, connections between this function and Euler totient function as well as with Riemann zeta function are established.