For each positive integer $m$, let $N(m)$ denote the number of $\varphi$-preimages of $m$, where $\varphi$ is Euler’s totient function. For example, $N(12) = 6$, corresponding to the six preimages 13, 21, 26, 28, 36, and 42. We discuss several statistical questions concerning $N(m)$ — for instance, its average size, its maximal order, and the typical size of
$N(\varphi(k))$ as $k$ varies.
Links:
[1] https://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/node/3444
[3] https://www.mpim-bonn.mpg.de/node/246