An Identity involving the Cyclotomic Polynomials

TL;DR

This paper introduces an elementary multiplicative identity for cyclotomic polynomials, connecting their properties with arithmetical functions, and discusses its prior appearance in existing literature.

Contribution

The paper presents a new elementary identity for cyclotomic polynomials and explores its implications and connections with arithmetical functions.

Findings

Identified a multiplicative property of cyclotomic polynomials

Connected the identity with arithmetical functions

Acknowledged prior similar results in literature

Abstract

We present an elementary identity for the cyclotomic polynomials $\Phi_n(X)$ which reflects a kind of multiplicative property of $\Phi_n(X)$ as a function of $n$, and we explore its connections with the properties of other arithmetical functions. Important Note: In the first version of this article uploaded to the arXiv, it is said that this result seemed to be new. However, after that, I have learned that this identity (in Theorem 1.1) has previously appeared in Cheng, C. C. A., McKay, J. H., & Wang, S. S. S. (1995). Resultants of cyclotomic polynomials. Proceedings of the American Mathematical Society, 123(4), 1053-1059. (with a different proof).