The Asymptotic Behavior of Compositions of the Euler and Carmichael Functions

TL;DR

This paper investigates the asymptotic properties of compositions involving Euler's totient and Carmichael's lambda functions, revealing their growth patterns and ratios in the limit.

Contribution

It provides a detailed analysis of the asymptotic behavior of compositions of the Euler and Carmichael functions, including the normal order of their ratios.

Findings

Determined the normal order of the logarithm of the ratio of compositions.

Compared the asymptotic behaviors of different compositions of the functions.

Established new results on the growth rates of these function compositions.

Abstract

We compare the asymptotic behavior of Carmichael's lambda function composed with Euler's totient function to the asymptotic behavior of Carmichael's lambda function composed with itself. We establish the normal order of the logarithm of the ratio.