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

Abstract

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 $g_{k,f}(n):=\frac{{\prod }_{i=0}^{k}f(b+a(n+ic))}{f({\mathrm{lcm}}_{0\le i\le k}\{b+a(n+ic)\})}$. We first show that $g_{k,f}$ is periodic and $c\mathrm{lcm}(1,...,k)$ is its period. Consequently, we provide a detailed local analysis to the periodic function $g_{k,\phi }$, and determine the smallest period of $g_{k,\phi }$, where $\phi$ is the Euler phi function.

16 pages. To appear in Journal of Number Theory