  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).
