The Maximal Order of Iterated Multiplicative Functions

Abstract

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 max n≤x log d(n) = log x log log x (log 2 + o (1)).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 log f (f (n)) for a class of multiplicative functions f . 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 f arising as a normalized function counting representations by certain binary quadratic forms. Incidentally, for the non-multiplicative function r2 which counts how often a positive integer is represented as a sum of two squares, this entails the asymptotic formula max n≤x log r2(r2(n)) = √ log x log log x (c/ √ 2 + o(1)) with some explicitly given constant c > 0.