The iterated Carmichael lambda function

The Asymptotic Behavior of Compositions of the Euler and Carmichael Functions

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.

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

The Carmichael lambda function $λ(n)$ is defined to be the smallest positive integer $m$ such that $a^m \equiv 1 \pmod{n}$ for all $(a,n)=1.$ $λ_k(n)$ is defined to be the $k$th iterate of $λ(n).$ Let L(n) be the smallest $k$ for which $λ_k(n)=1.$ 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 $L(n) \ll (\log n)^δ$ for almost all $n,$ for some $δ<1.$ We will also show $L(n) \gg \log\log n$ for almost all $n$ and conjecture a normal order for L(n).

Radically weakening the Lehmer and Carmichael conditions

Lehmer's totient problem asks if there exist composite integers n satisfying the condition phi(n)|(n-1), (where phi is the Euler-phi function) while Carmichael numbers satisfy the weaker condition lambda(n)|(n-1) (where lambda is the Carmichael universal exponent function). We weaken the condition further, looking at those composite n where each prime divisor of phi(n) also divides n-1. (So rad(phi(n))|(n-1).) While these numbers appear to be far more numerous than the Carmichael numbers, we show that their distribution has the same rough upper bound as that of the Carmichael numbers, a bound …

Two problems on the distribution of Carmichael's lambda function

Let $λ(n)$ denote the exponent of the multiplicative group modulo $n$. We show that when $q$ is odd, each coprime residue class modulo $q$ is hit equally often by $λ(n)$ as $n$ varies. Under the stronger assumption that $\gcd(q,6)=1$, we prove that equidistribution persists throughout a Siegel--Walfisz-type range of uniformity. By similar methods we show that $λ(n)$ obeys Benford's leading digit law with respect to natural density. Moreover, if we assume GRH, then Benford's law holds for the order of $a$ mod $n$, for any fixed integer $a\notin \{0,\pm 1\}$.

The iterated Carmichael λ-function and the number of cycles of the power generator

Iteration of the modular l-th power function f(x) = x^l (mod n) provides a common pseudorandom number generator (known as the Blum-Blum-Shub generator when l=2). The period of this pseudorandom number generator is closely related to λ(λ(n)), where λ(n) denotes Carmichael's function, namely the maximal multiplicative order of any integer modulo n. In this paper, we show that for almost all n, the size of λ(λ(n)) is n/exp((1+o(1))(log log n)^2 log log log n). We conjecture an analogous formula for the k-th iterate of λ. We deduce that for almost all n, the psuedorandom number generator described…

Carmichael λ函数的小值

Small values of Carmichael's lambda function

研究λ(n)的分布，给出L(x,y)的上界，并在合理假设下渐近精确，应用于2模n的阶远小于x^{1/2}的奇数n的计数。