THE MAXIMAL ORDER OF ITERATED MULTIPLICATIVE FUNCTIONS

Following Wigert, various authors, including Ramanujan, Gronwall, Erdős, Ivić, Schwarz, Wirsing and Shiu, determined the maximal order of several multiplicative functions, generalizing Wigert's result (Formula presented.) On the contrary, for many multiplicative functions, the maximal order of iterations of the functions remains widely open. The case of the iterated divisor function was only solved recently, answering a question of Ramanujan from 1915. Here we determine the maximal order of (Formula presented.) for a class of multiplicative functions (Formula presented.). In particular, this class contains functions counting ideals of given norm in the ring of integers of an arbitrary, fixed quadratic number field. As a consequence, we determine such maximal orders for several multiplicative (Formula presented.) arising as a normalized function counting representations by certain binary quadratic forms. Incidentally, for the non-multiplicative function (Formula presented.) which counts how often a positive integer is represented as a sum of two squares, this entails the asymptotic formula (Formula presented.) with some explicitly given constant (Formula presented.).