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

Let $k\ge 1,a\ge 1,b\ge 0$ and $c\ge 1$ be integers. Let $f$ be a multiplicative function with $f(n)\ne 0$ for all positive integers $n$. We define the arithmetic function $g_{k,f}$ for any positive integer $n$ by $g_{k,f}(n):=\frac{\prod_{i=0}^k f(b+a(n+ic))} {f({\rm lcm}_{0\le i\le k} \{b+a(n+ic)\})}$. We first show that $g_{k,f}$ is periodic and $c {\rm lcm}(1,...,k)$ is its period. Consequently, we provide a detailed local analysis to the periodic function $g_{k,\varphi}$, and determine the smallest period of $g_{k,\varphi}$, where $\varphi$ is the Euler phi function.