On Arithmetic Functions Related to Iterates of the Schemmel Totient Functions

Abstract

We begin by introducing an interesting class of functions, known as the Schemmel totient functions, that generalizes the Euler totient function. For each Schemmel totient function $L_m$, we define two new functions, denoted $R_m$ and $H_m$, that arise from iterating $L_m$. Roughly speaking, $R_m$ counts the number of iterations of $L_m$ needed to reach either $0$ or $1$, and $H_m$ takes the value (either $0$ or $1$) that the iteration trajectory eventually reaches. Our first major result is a proof that, for any positive integer $m$, the function $H_m$ is completely multiplicative. We then introduce an iterate summatory function, denoted $D_m$, and define the terms $D_m$-deficient, $D_m$-perfect, and $D_m$-abundant. We proceed to prove several results related to these definitions, culminating in a proof that, for all positive even integers $m$, there are infinitely many $D_m$-abundant numbers. Many open problems arise from the introduction of these functions and terms, and we mention a few of them, as well as some numerical results.