### 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.