The maximal order of iterated multiplicative functions

TL;DR

This paper determines the maximal order of the iterated logarithm of a class of multiplicative functions, including those counting ideals in quadratic fields and representations by quadratic forms, extending classical results on multiplicative functions.

Contribution

It establishes the maximal order of \\log f(f(n)) for a broad class of multiplicative functions, including those related to quadratic number fields and quadratic forms.

Findings

Maximal order of \\log f(f(n)) determined for a class of multiplicative functions.

Includes functions counting ideals in quadratic number fields.

Provides asymptotic formula for \\max_{n \\leq x} \\log r_2(r_2(n)).

Abstract

Following Wigert, various authors, including Ramanujan, Gronwall, Erd\H{o}s, Ivi\'{c}, Schwarz, Wirsing, and Shiu, determined the maximal order of several multiplicative functions, generalizing Wigert's result $$\max_{n\leq x} \log d(n) = \frac{\log x}{\log \log x} (\log 2 + o(1)). $$ 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 $r_2$ which counts how often a positive integer is represented as a sum of two squares, this entails the asymptotic formula $$ \max_{n\leq x} \log r_2(r_2(n))= \frac{\sqrt{\log x}}{\log \log x} (c/\sqrt{2}+o(1))$$ with some explicitly given constant $c>0$.