The number of iterates of the Carmichael lambda function required to reach 1

TL;DR

This paper investigates the iterated Carmichael lambda function, establishing bounds on the number of iterations needed to reach 1 for almost all integers, and proposes a conjecture on its typical behavior.

Contribution

It proves that for almost all n, the number of iterations L(n) is bounded above by a power of log n with exponent less than 1, and provides lower bounds and conjectures on its normal order.

Findings

L(n)  (\u03bb log n)^{} for some <1 for almost all n

L(n)   log log n for almost all n

Conjecture on the normal order of L(n)

Abstract

The Carmichael lambda function $\lambda(n)$ is defined to be the smallest positive integer $m$ such that $a^m \equiv 1 \pmod{n}$ for all $(a,n)=1.$ $\lambda_k(n)$ is defined to be the $k$th iterate of $\lambda(n).$ Let L(n) be the smallest $k$ for which $\lambda_k(n)=1.$ It's easy to show that $L(n) \ll \log n.$ It's conjectured that $L(n)\asymp \log\log n,$ but previously it was not known to be $o(\log n)$ for almost all $n.$ We will show that $L(n) \ll (\log n)^{\delta}$ for almost all $n,$ for some $\delta <1.$ We will also show $L(n) \gg \log\log n$ for almost all $n$ and conjecture a normal order for L(n).