On Arithmetic Functions Related to Iterates of the Schemmel Totient Functions
Colin Defant

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 , we define two new functions, denoted and , that arise from iterating . Roughly speaking, counts the number of iterations of needed to reach either or , and takes the value (either or ) that the iteration trajectory eventually reaches. Our first major result is a proof that, for any positive integer , the function is completely multiplicative. We then introduce an iterate summatory function, denoted , and define the terms -deficient, -perfect, and -abundant. We proceed to prove several results related to these definitions, culminating in a proof that, for all positive even integers , there are infinitely many -abundant…
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Taxonomy
TopicsAnalytic Number Theory Research · Advanced Mathematical Identities · History and Theory of Mathematics
