On the periodicity of a class of arithmetic functions associated with multiplicative functions

Let k⩾1, a⩾1, b⩾0 and c⩾1 be integers. Let f be a multiplicative function with f(n)≠0 for all positive integers n. We define the arithmetic function gk,f for any positive integer n by . We first show that gk,f is periodic and clcm(1,…,k) is its period. Consequently, we provide a detailed local analysis to the periodic function gk,φ, and determine the smallest period of gk,φ, where φ is the Euler phi function.