On the periodicity of some Farhi arithmetical functions

TL;DR

This paper proves that a class of Farhi arithmetic functions, defined via polynomials with coprimality conditions, are periodic, extending previous results by Farhi, Kane, Hong, and Yang.

Contribution

It establishes the periodicity of Farhi arithmetic functions for a broad class of polynomials, generalizing earlier specific cases.

Findings

Proves the periodicity of g_{k,f} functions.

Generalizes previous results to broader polynomial classes.

Extends understanding of arithmetic function periodicity.

Abstract

Let $k\in\mathbb{N}$. Let $f(x)\in \Bbb{Z}[x]$ be any polynomial such that $f(x)$ and $f(x+1)f(x+2)... f(x+k)$ are coprime in $\mathbb{Q}[x]$. We call $$g_{k,f}(n):=\frac{|f(n)f(n+1)... f(n+k)|}{\text{lcm}(f(n),f(n+1),...,f(n+k))}$$ a Farhi arithmetic function. In this paper, we prove that $g_{k,f}$ is periodic. This generalizes the previous results of Farhi and Kane, and Hong and Yang.