Extremal orders of compositions of certain arithmetical functions

TL;DR

This paper investigates the extremal growth rates of compositions of specific arithmetical functions like divisor sums and Euler's totient, providing comprehensive generalizations and refinements of existing results.

Contribution

It offers a complete analysis of the extremal orders of compositions of arithmetical functions, extending and refining prior findings in the field.

Findings

Determined exact extremal orders for compositions of divisor sum and Euler functions.

Generalized known extremal order results to broader classes of arithmetical functions.

Refined bounds and asymptotic behaviors for these compositions.

Abstract

We study the exact extremal orders of compositions $f(g(n))$ of certain arithmetical functions, including the functions $\sigma(n)$, $\phi(n)$, $\sigma^*(n)$ and $\phi^*(n)$, representing the sum of divisors of $n$, Euler's function and their unitary analogues, respectively. Our results complete, generalize and refine known results.