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THE MAXIMAL ORDER OF ITERATED MULTIPLICATIVE FUNCTIONS

Part of: Multiplicative number theory

Published online by Cambridge University Press:  30 July 2019

Christian Elsholtz ,
Marc Technau  and
Niclas Technau
Christian Elsholtz
Affiliation:
Institute of Analysis and Number Theory, Graz University of Technology, Kopernikusgasse 24, 8010 Graz, Austria email elsholtz@math.tugraz.at
Marc Technau
Affiliation:
Institute of Analysis and Number Theory, Graz University of Technology, Kopernikusgasse 24, 8010 Graz, Austria email mtechnau@math.tugraz.at
Niclas Technau
Affiliation:
Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, 69978 Tel Aviv, Israel email niclast@mail.tau.ac.il
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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

$$\begin{eqnarray}\max _{n\leqslant x}\log d(n)=\frac{\log x}{\log \log x}(\log 2+o(1)).\end{eqnarray}$$
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
$$\begin{eqnarray}\max _{n\leqslant x}\log r_{2}(r_{2}(n))=\frac{\sqrt{\log x}}{\log \log x}(c/\sqrt{2}+o(1))\end{eqnarray}$$
 with some explicitly given constant $c>0$.

MSC classification

Primary: 11N37: Asymptotic results on arithmetic functions
Type
Research Article
Information
Mathematika , Volume 65 , Issue 4 , 2019 , pp. 990 - 1009
Copyright
Copyright © University College London 2019 

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Footnotes

The third author was supported, while working on this paper, by the Austrian Science Fund (FWF): by either Project W1230 Doctoral Program “Discrete Mathematics” or Project Y-901.

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