The iterated Carmichael lambda function

TL;DR

This paper investigates the iterated Carmichael lambda function, establishing a normal order for the ratio of n to its k-th iterate for all k, extending previous results for k=1,2.

Contribution

It provides a generalization of the normal order of n/λ_k(n) for all iterates k, advancing understanding of the Carmichael lambda function's behavior.

Findings

Normal order for n/λ_k(n) established for all k

Extends previous results limited to k=1,2

Provides insights into the iterated Carmichael lambda function

Abstract

The Carmichael lambda function $\lambda(n)$ is defined to be the smallest positive integer $m$ such that $a^m$ is congruent to 1 modulo $n,$ for all $a$ and $n$ relatively prime. The function $\lambda_k(n)$ is defined to be the $k$th iterate of $\lambda(n).$ Previous results show a normal order for $n/\lambda_k(n)$ where $k=1,2.$ We will show a normal order for all $k.$