Jordan totient quotients

TL;DR

This paper investigates the average behavior of Jordan totient quotients, introducing two methods to analyze these functions and applying them to derive results about cyclotomic polynomials and their derivatives.

Contribution

It presents two general methods for analyzing Jordan totient quotients and applies them to various problems involving cyclotomic polynomials and their derivatives.

Findings

Average behavior of Jordan totient quotients determined.

Derived the average order of derivatives of cyclotomic polynomials.

Analyzed the Schwarzian derivative of cyclotomic polynomials.

Abstract

The Jordan totient $J_k(n)$ can be defined by $J_k(n)=n^k\prod_{p\mid n}(1-p^{-k})$. In this paper, we study the average behavior of fractions $P/Q$ of two products $P$ and $Q$ of Jordan totients, which we call Jordan totient quotients. To this end, we describe two general and ready-to-use methods that allow one to deal with a larger class of totient functions. The first one is elementary and the second one uses an advanced method due to Balakrishnan and P\'etermann. As an application, we determine the average behavior of the Jordan totient quotient, the $k^{th}$ normalized derivative of the $n^{th}$ cyclotomic polynomial $\Phi_n(z)$ at $z=1$, the second normalized derivative of the $n^{th}$ cyclotomic polynomial $\Phi_n(z)$ at $z=-1$, and the average order of the Schwarzian derivative of $\Phi_n(z)$ at $z=1$.