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

TL;DR

This paper investigates the periodicity of a class of arithmetic functions derived from multiplicative functions, establishing their periods and analyzing specific cases involving the Euler phi function.

Contribution

It proves that the defined functions are periodic with explicitly determined periods and provides a detailed analysis of the smallest period for functions involving Euler's phi function.

Findings

g_{k,f} is periodic with period c * lcm(1,...,k)

The smallest period of g_{k,φ} is explicitly determined

Periodic structure depends on the parameters and the multiplicative function

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 $g_{k,f}(n):=\frac{\prod_{i=0}^k f(b+a(n+ic))} {f({\rm lcm}_{0\le i\le k} \{b+a(n+ic)\})}$. 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.