On Arithmetic Functions Related to Iterates of the Schemmel Totient Functions

TL;DR

This paper introduces new functions derived from Schemmel totient functions, explores their properties, and proves the infinitude of certain abundant numbers for even parameters, expanding understanding of arithmetic functions related to iterates.

Contribution

It defines and analyzes new functions from Schemmel totient functions, proving properties like complete multiplicativity and the infinitude of abundant numbers for even parameters.

Findings

H_m is completely multiplicative for all positive m

D_m-abundant numbers are infinite for even m

Introduces new classifications of numbers based on D_m 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.