I learned that from here for Euler totient function $\phi (n)$ , we have

$$\lim_{n\to \infty}\text{sup} \frac{\phi (n)}{n}=1$$

$$\lim_{n\to \infty}\text{inf} \frac{\phi (n)}{n}=0$$

However, I could not find such a limit for the Carmichael function $\lambda (n)$ which is associated with the Euler totient function. So I'm curious about the following limits:

$$\lim_{n\to \infty}\text{sup} \frac{\lambda (n)}{n}$$ and $$\lim_{n\to \infty}\text{inf} \frac{\lambda (n)}{n}$$

I think for any $n\in\mathbb{Z^{+}}$ we have $\phi(n)≥\lambda(n)$. So, are the following limits correct?

$\lim_{n\to \infty}\text{sup} \frac{\lambda (n)}{n}=1$ and $\lim_{n\to \infty}\text{inf} \frac{\lambda (n)}{n}=0$

I could not find these limits.

Thank you.

  • 2
    $\begingroup$ The limsup for the totient is from the sequence of primes. The same holds with Carmichael function, because $\lambda(p)=\phi(p)=p-1$ for primes $p$. The liminf is due to $\lambda(n)\leq \phi(n)$, we have $\liminf (\lambda(n)/n )\leq \liminf (\phi(n)/n)$. This is from the sequence of primorials. $\endgroup$ – i707107 Jan 27 at 18:43
  • $\begingroup$ @i707107 thank you for comment. You mean $ \liminf (\lambda(n)/n )\leq \liminf (\phi(n)/n) \Rightarrow \liminf (\lambda(n)/n )=0 $ isn't it? $\endgroup$ – Elementary Jan 27 at 19:00
  • 1
    $\begingroup$ That is correct. $\endgroup$ – i707107 Jan 27 at 19:00
  • $\begingroup$ @i707107 Then, What can we say about $\limsup (\lambda(n)/n ) ?$ $\endgroup$ – Elementary Jan 27 at 19:03
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    $\begingroup$ If your problem is, in the end, to understand a proof that $\lambda(p)=p-1$ for every prime number $p$, you might want to post a question asking this. $\endgroup$ – Did Jan 28 at 11:56

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