Title: The maximal order of iterated multiplicative functions,
Abstract: Following Wigert, a great number of authors including Ramanujan, Gronwall,
Erdős, Ivić, Heppner, J. Knopfmacher, Nicolas, Schwarz, Wirsing,
Freiman, Shiu et al. determined the maximal order of several multiplicative
functions, generalizing Wigert's result \[\max_{n\leq x} \log d(n)= (\log
2+o(1))\frac{\log x}{\log \log x}.\]
On the contrary, for many multiplicative functions, the maximal order of
iterations of the functions remains wide open. The case of the iterated divisor
function was only recently solved, answering a question of Ramanujan (1915).
Here, we determine the maximal order of $\log f(f(n))$ for a class of
multiplicative functions $f$ which are related to the divisor function. As a
corollary, we apply this to the function counting representations as sums of
two squares of non-negative integers, also known as $r_2(n)/4$, and obtain an
asymptotic formula: \[\max_{n\leq x} \log f(f(n))= (c+o(1))\frac{\sqrt{\log
x}}{\log \log x},\] with some explicitly given positive constant $c$.