Smarandache Type Function Obtained by Duality

In this paper we extend the Smarandache function from the set $N\ast$ 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.