The maximal order of iterated multiplicative functions

Christian Elsholtz, Marc Technau, Niclas Technau

Published 2017-09-14Version 1

Following Wigert, a great number of authors including Ramanujan, Gronwall, Erd\H{o}s, Ivi\'{c}, 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 $c$.