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Mathematics, Number Theory
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 $z=1$.
