### 29 maj, kl 11.00-11.50, KTH, sal 3721.

###
Florian Luca (Universidad Nacional Autonoma de Mexico):
On the iterates of the Euler function

**Abstract. (Here as pdf.)**
Let $\phi(n)$ be the Euler function of the positive
integer $n$. For a positive integer $k$, let $\phi^{(k)}(n)$ be the
$k$th fold iterate of the function $\phi(n)$. In my talk, I will
look at the range of the function $\phi^{(k)}(n)$. For example,
putting $V_k(x)=\#\{\phi^{(k)}(n)\le x\}$, then for $x$ sufficiently
large the estimate

$$
\#V_k(x)\le {{x}\over {(\log x)^k}}\exp(13k^{3/2}(\log\log
x\log\log\log x)^{1/2})
$$

holds uniformly in $k\ge 1$. Under the prime $k$-tuples conjecture I
show that $\#V_k(x)\gg_k x/(\log x)^k$. I will also give the main
ideas of an unconditional proof of this lower bound when $k=2$.
These results have been obtained jointly with Carl Pomerance.