Following Wigert, various authors, including Ramanujan, Gronwall, Erdős, Ivic, 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 $c>0$.

中文翻译：

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迭代乘法函数的最大阶数

在 Wigert 之后，包括 Ramanujan、Gronwall、Erdős、Ivic、Schwarz、Wirsing 和 Shiu 在内的多位作者确定了几个乘法函数的最大阶数，概括了 Wigert 的结果 $$\max_{n\leq x} \log d(n) = \frac{\log x}{\log \log x} (\log 2 + o(1))。$$ 相反，对于许多乘法函数，函数迭代的最大阶数仍然广泛开放。迭代除数函数的情况最近才解决，回答了1915年Ramanujan的一个问题。这里我们确定一类乘法函数$f$的$\log f(f(n))$的最大阶数。特别是，此类包含对任意固定二次数域的整数环中给定范数的理想进行计数的函数。作为结果，我们确定几个乘法 $f$ 的最大阶数，作为归一化函数计数表示的某些二进制二次形式。顺便说一句，对于计算正整数表示为两个平方和的频率的非乘法函数 $r_2$，这需要渐近公式 $$ \max_{n\leq x} \log r_2(r_2(n) )= \frac{\sqrt{\log x}}{\log \log x} (c/\sqrt{2}+o(1))$$ 带有一些明确给出的常量 $c>0$。