\documentclass[12pt,reqno]{article} \usepackage[usenames]{color} \usepackage{amssymb} \usepackage{graphicx} \usepackage{amscd} \usepackage[colorlinks=true, linkcolor=webgreen, filecolor=webbrown, citecolor=webgreen]{hyperref} \definecolor{webgreen}{rgb}{0,.5,0} \definecolor{webbrown}{rgb}{.6,0,0} \usepackage{color} \usepackage{fullpage} \usepackage{float} \usepackage{psfig} \usepackage{graphics,amsmath,amssymb} \usepackage{amsthm} \usepackage{amsfonts} \usepackage{latexsym} \usepackage{epsf} \usepackage{enumerate} \setlength{\textwidth}{6.5in} \setlength{\oddsidemargin}{.1in} \setlength{\evensidemargin}{.1in} \setlength{\topmargin}{-.1in} \setlength{\textheight}{8.4in} \DeclareMathOperator\Rea{Re} \newcommand{\seqnum}[1]{\href{http://oeis.org/#1}{\underline{#1}}} \begin{document} \begin{center} \epsfxsize=4in \leavevmode\epsffile{logo129.eps} \end{center} \theoremstyle{plain} \newtheorem{theorem}{Theorem} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{conjecture}[theorem]{Conjecture} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \begin{center} \vskip 1cm{\LARGE\bf A Generalization of the Gcd-Sum Function } \vskip 1cm \large Ulrich Abel, Waqar Awan, and Vitaliy Kushnirevych \\ Fachbereich MND\\ Technische Hochschule Mittelhessen\\ University of Applied Sciences\\ Wilhelm-Leuschner-Stra\ss{}e 13 \\ 61169 Friedberg\\ Germany\\ \href{mailto:Ulrich.Abel@mnd.thm.de}{\tt Ulrich.Abel@mnd.thm.de}\\ \href{mailto:Waqar.Awan@stud.h-da.de}{\tt Waqar.Awan@stud.h-da.de}\\ \href{mailto:Vitaliy.Kushnirevych@mnd.thm.de}{\tt Vitaliy.Kushnirevych@mnd.thm.de} \end{center} \vskip .2 in \newcommand{\Prod}{\displaystyle\prod\limits} \newcommand{\Sum}{\displaystyle\sum\limits} \begin{abstract} In this paper we consider the generalization $G_{d}(n)$ of the Broughan gcd-sum function, i.e., the sum of such gcd's that are divisors of the positive integer $d$. Examples of Dirichlet series and asymptotic relations for $G_{d}$ and related functions are given. \end{abstract} \section{Introduction} In the recent article \cite{Broughan-2001}, Broughan studies the sum of the greatest common divisors of the first $n$ positive integers with $n$, i.e., the arithmetic function \begin{equation*} G(n):=\Sum_{k=1}^{n}\gcd (k,n). \end{equation*} This function arises in deriving asymptotic estimates for a lattice point counting problem \cite[Sect.\ 5]{Broughan-2001}. The function $G$ has polynomial growth as $n$ tends to infinity. For $p\in \mathbb{P}$ (throughout the paper $\mathbb{P}$ denotes the set of prime numbers) and $\alpha \in \mathbb{N}$, it is not difficult to show that \begin{equation*} G(p^{\alpha })=\Sum_{j=0}^{\alpha -1}\underbrace{(p-1)p^{\alpha -1-j}}_{\text{number of $\gcd $'s equal to $p^{j}$}}p^{j}+1\cdot p^{\alpha }=(\alpha +1)p^{\alpha }-\alpha p^{\alpha -1}. \end{equation*} (cf.\ \cite[Th.\ 2.2]{Broughan-2001}). Following \cite[Cor.\ 2.1]{Broughan-2001} $G$ is a multiplicative function, i.e., $G(mn)=G(m)G(n)$ for coprime $m,n\in \mathbb{N}$, that is, $\gcd (m,n)=1$. The corresponding Dirichlet series $\mathcal{G}\left( s\right) $ converges at all points of the complex plane, except at the zeros of the Riemann zeta function and the point $s=2$, where it has a double pole. Moreover, Broughan derives asymptotic expressions for the partial sums of the Dirichlet series at all real values of $s$. The following generalization of $G$ (see \cite{Awan-2012}) arises in the study of distribution of determinant values in residue class rings. For $d\in \mathbb{N}$, we introduce the function \begin{equation*} G_{d}(n):=\Sum_{\substack{ k=1 \\ \gcd (k,n)\mid d}}^{n}\gcd (k,n). \end{equation*} Obviously, $G(n)=G_{n}(n)$ and $G_{1}(n)=\varphi (n)$, where $\varphi $ is Euler's totient function. The purpose of this note is to study the function $G_{d}$. In the next section we present some elementary properties of $G_{d}$. Furthermore, we study the corresponding Dirichlet series $\mathcal{G}_{d}\left( s\right)$. Some of the results will be applied in a forthcoming paper on the distribution of determinant values in residue class rings and finite fields. As an example we mention that in the residue class ring $\mathbb{Z}_{n}$ $\left( n\in \mathbb{N}\right) $, for $r\in \mathbb{Z}_{n}$, \begin{equation*} H_{n}(r)=|\{(i,j)\in \mathbb{Z}_{n}\times \mathbb{Z}_{n}\mid i\cdot j=r\}|, \end{equation*} the number of products equal to $r$ having precisely two factors in $\mathbb{Z}_{n}$, is equal to \begin{equation*} H_{n}(r)= \begin{cases} G_{n}(n)=G(n), & \text{if $r=0$}; \\ G_{d}(n)=G_{\gcd (r,n)}(n), & \text{if $r\neq 0$}. \end{cases} \end{equation*} A similar problem as the calculation of the value $H_n(r)$ in the domain of positive integers is the so-called multiplication table problem posed by Erd\H{o}s (see \cite{Erdos1955}): how many integers can be written as a product $i\cdot j$ for a given positive integer $n\in\mathbb N$ with positive integers $i\leqslant n$ and $j\leqslant n$? Erd\H{o}s (\cite{Erdos1955, Erdos1960}) gave the first estimates of this quantity. Tenenbaum \cite{Tenenbaum} had made the results of Erd\H{o}s more precise. Ford (\cite{Ford2008-1, Ford2008-2}) derived the exact order of magnitude of the $n\times n$ multiplication table size completely. Koukoulopolous \cite{Kou2010-1, Kou2013-2} presents a perfect overview of the actual situation and the further development of Ford-Erd\H{o}s results. \section{Properties of \texorpdfstring{$G_d$}{}} The following lemma gathers some elementary properties of $G_{d}(n)$. \begin{lemma} \label{lemma1}~ \begin{enumerate} \item[(i)] \label{erste-eigenschaft} For $m,n\in \mathbb{N}$, we have $G_{m}(n)=G_{\gcd (m,n)}(n)$.\newline In particular, for $m,\alpha \in \mathbb{N}$, $p\in \mathbb{P}$, we have $G_{\gcd \left( m,p^{\alpha }\right) }(p^{\alpha })=G_{m}(p^{\alpha })$; \item[(ii)] \label{Gd-und-euler} for coprime $d,n\in \mathbb{N}$, we have $G_{d}(n)=\varphi (n)$; \item[(iii)] \label{G-d1d2} for $d=d_{1}d_{2}$ with $\gcd (d_{1},n)=1$, we have $G_{d}(n)=G_{d_{2}}(n)$. \end{enumerate} \end{lemma} \begin{proof} \begin{enumerate}[(i)] \item Since $\gcd (k,n)\mid \gcd (m,n)\iff \gcd (k,n)\mid m$ for all $m,n,k\in \mathbb{N}$, the first formula follows from the definition. One obtains the second one by substituting $n=p^{\alpha }$. \item If $\gcd (d,n)=1$, using (i) we get \begin{equation*} G_{d}(n)=G_{1}(n)=\varphi (n). \end{equation*} \item Since $\gcd (d_{1},n)=1\Rightarrow \gcd (d_{1}d_{2},n)=\gcd (d_{2},n)$, it follows that \begin{equation*} G_{d}(n)=G_{d_{1}d_{2}}(n)=G_{\gcd (d_{1}d_{2},n)}(n)=G_{\gcd (d_{2},n)}(n)=G_{d_{2}}(n), \end{equation*} where we used (i) twice. \end{enumerate} The proof of the lemma is completed. \end{proof} Let $\rho _{d}$ denote the multiplicative function \begin{equation*} \rho _{d}(w)= \begin{cases} w, & \text{if } w\mid d; \\ 0, & \text{if } w\nmid d. \end{cases} \end{equation*} Then we have the representation \begin{equation} G_{d}=\rho _{d}\ast \varphi , \label{gd=rho*phi} \end{equation} where $\ast $ denotes Dirichlet product. Indeed, \begin{equation*} G_{d}(n)=\Sum_{\substack{ k=1 \\ \gcd (k,n)\mid d}} ^{n}\gcd (k,n)=\Sum_{\substack{ w\mid d \\ w\mid n}} w\varphi \left( \frac{n}{w}\right) =\Sum_{w\mid n}\rho _{d}(w)\varphi \left( \frac{n}{w}\right) =(\rho _{d}\ast \varphi )(n). \end{equation*} Therefore, $G_{d}$ is multiplicative as it is the Dirichlet product of multiplicative functions \cite[Th.\ 2.5(c) and Th.\ 2.14]{Apostol}. \begin{theorem} \label{Gd-mult}$G_{d}$ is a multiplicative function, i.e., for coprime $m,n\in \mathbb{N}$, we have \begin{equation*} G_{d}(mn)=G_{d}(m)G_{d}(n). \end{equation*} \end{theorem} We also give a direct proof of the preceding theorem. \begin{proof} \label{Gd-mult-beweis}Let $d\mid n_{1}n_{2}$ with coprime $n_{1},n_{2}\in \mathbb{N}$. This implies $d=d_{1}d_{2}$ with $d_{1}\mid n_{1}$ and $d_{2}\mid n_{2}$, so that $d_{1}$ and $d_{2}$ are coprime. One has \begin{equation*} G_{d}(n_{1}n_{2})=G_{d_{1}d_{2}}(n_{1}n_{2})= \Sum_{w\mid d_{1}d_{2}}w\varphi \left( \frac{n_{1}n_{2}}{w}\right) =\Sum_{w_{1}\mid d_{1}}\Sum_{w_{2}\mid d_{2}}w_{1}w_{2}\varphi \left( \frac{n_{1}n_{2}}{w_{1}w_{2}}\right). \end{equation*} Because $\varphi $ is multiplicative and $\gcd \left( \frac{n_{1}}{w_{1}}, \frac{n_{2}}{w_{2}}\right) =1$, one obtains \begin{equation*} G_{d}(n_{1}n_{2}) =\Sum_{w_{1}\mid d_{1}}w_{1}\varphi \left( \frac{n_{1}}{w_{1}}\right) \Sum_{w_{2}\mid d_{2}}w_{2}\varphi \left( \frac{n_{2}}{w_{2}}\right) = G_{d_{1}}\left( n_{1}\right) G_{d_{2}}\left( n_{2}\right) =G_{d}\left( n_{1}\right) G_{d}\left( n_{2}\right) . \end{equation*} This completes the proof. \end{proof} \begin{theorem} \label{Gd-d-mult} For $n\in \mathbb{N}$ and for coprime $d_{1},d_{2}\in \mathbb{N}$, we have \begin{equation*} G_{d_{1}}(n)\cdot G_{d_{2}}(n)=\varphi (n)\cdot G_{d_{1}d_{2}}(n). \end{equation*} In particular, $G_{d_{1}d_{2}}(n)\mid G_{d_{1}}(n)G_{d_{2}}(n)$. \end{theorem} \begin{proof} Let $d=d_{1}d_{2}$ with $\gcd (d_{1},d_{2})=1$. By Equation \eqref{gd=rho*phi} we have \begin{equation*} G_{d}(n)=(\rho_{d}\ast\varphi )(n)=\Sum_{w\mid n}\rho_{d_{1}d_{2}}(w)\varphi\left( \frac{n}{w}\right) =\Sum_{w_{1}\mid n}\Sum_{w_{2}\mid n}\rho_{d_{1}}(w_{1})\rho _{d_{2}}(w_{2})\varphi \left( \frac{n}{w_{1}w_{2}} \right). \end{equation*} Now, decompose $n=kn_{1}n_{2}$ in a product of three pairwise coprime factors $k$, $n_{1}$, $n_{2}$ such that $d_{i}\mid n_{i}$ $\left( i=1,2\right) $. If $w_{i}\mid d_{i}$ $\left( i=1,2\right) $ we conclude that \begin{equation*} \varphi \left( \frac{n}{w_{1}w_{2}}\right) =\varphi (k)\varphi \left( \frac{n_{1}}{w_{1}}\right) \varphi \left( \frac{n_{2}}{w_{2}}\right) =\varphi (k)\;\dfrac{\varphi \left( \frac{n}{w_{1}}\right) \varphi \left( \frac{n}{w_{2}}\right) }{\varphi (kn_{2})\varphi (kn_{1})}=\dfrac{\varphi \left( \frac{n}{w_{1}}\right) \varphi \left( \frac{n}{w_{2}}\right)}{\varphi (n)}. \end{equation*} Hence, we obtain \begin{equation*} \varphi \left( n\right) G_{d}(n)=\Sum_{w_{1}\mid n}\rho _{d_{1}}(w_{1})\varphi \left( \frac{n}{w_{1}}\right) \Sum_{w_{2}\mid n}\rho _{d_{2}}(w_{2})\varphi \left( \frac{n}{w_{2}}\right) =G_{d_{1}}(n)\cdot G_{d_{2}}(n) \end{equation*} which is the desired formula. \end{proof} We close this section with the following nice formula. \begin{theorem} For all $n\in \mathbb{N}$, we have \begin{equation*} \Sum_{i=1}^{n}G_{i}(n)=n^{2}. \end{equation*} \end{theorem} \begin{proof} Analogously to the proof of Theorem~\ref{Gd-mult} one has \begin{align*} \Sum_{i=1}^{n}G_{i}(n)& = \Sum_{i=1}^{n}\Sum_{w\mid n}\rho _{i}(w)\varphi \left( \frac{n}{w}\right) =\Sum_{w\mid n}\varphi \left( \frac{n}{w}\right) \Sum_{i=1}^{n}\rho _{i}(w) \\ & =\Sum_{w\mid n}\varphi \left( \frac{n}{w}\right) w \Sum_{\substack{ 1\leqslant i\leqslant n \\ w\mid i}}1= \Sum_{w\mid n}\varphi \left( \frac{n}{w}\right) w\frac{n}{w}=n\Sum_{w\mid n}\varphi \left( w\right) =n^{2}, \end{align*} where we used that $\Sum_{w\mid n}\varphi \left( w\right) =n$. \end{proof} \section{Evaluation of \texorpdfstring{$G_d$}{} at positive integers} In this section we consider the problem how to calculate the values of $G_{d}(n)$ for positive integers. We start with the special case of prime powers. In the following $\delta _{\alpha \beta }$ denotes the Kronecker symbol defined by $\delta_{\alpha\beta}=\begin{cases} 1, &\text{if $\alpha=\beta$};\\ 0, &\text{otherwise}.\end{cases}$ \begin{proposition} \label{prop-Gd-fuer-palpha} For prime powers $n=p^{\alpha }$ ($\alpha \in \mathbb{N}$) and $d=p^{\beta }$, $\beta \leqslant \alpha $ ($\beta \in \mathbb{N}\cup \{0\}$), we have \begin{equation*} G_{p^{\beta }}(p^{\alpha })=\varphi (p^{\alpha })\left( 1+\beta +\dfrac{\delta _{\alpha \beta }}{p-1}\right). \end{equation*} For prime powers $n=p^{\alpha }$ ($\alpha \in \mathbb{N}$) and $d=p^{\beta }$, $\beta >\alpha $ ($\beta \in \mathbb{N}$), we have \begin{equation*} G_{p^{\beta }}(p^{\alpha })=G_{p^{\alpha }}(p^{\alpha })=\varphi (p^{\alpha })\left( 1+\alpha +\dfrac{1}{p-1}\right). \end{equation*} \end{proposition} \begin{proof} For $0<\beta <\alpha $, we have \begin{equation*} G_{p^{\beta }}(p^{\alpha })=\Sum_{j=0}^{\beta }(p^{\alpha -j}-p^{\alpha -j-1})p^{j}=(p^{\alpha }-p^{\alpha -1})(1+\beta )=\varphi (p^{\alpha })(1+\beta ), \end{equation*} and, for $\beta =\alpha $, \begin{align*} G_{p^{\beta }}(p^{\alpha })& =G_{p^{\alpha }}(p^{\alpha })=G(p^{\alpha })=(\alpha +1)p^{\alpha }-\alpha p^{\alpha -1} \\ & =(\alpha +1)(p^{\alpha }-p^{\alpha -1})+p^{\alpha -1}=\varphi (p^{\alpha })(1+\beta )+p^{\alpha -1}. \end{align*} In the case $\beta =0$ application of Lemma~\ref{lemma1} (ii) leads to $ G_{1}(p^{\alpha })=\varphi (p^{\alpha })=\varphi (p^{\alpha })(1+\beta )$. Thus, for all $0\leqslant \beta \leqslant \alpha $, one has \begin{equation*} G_{p^{\beta }}(p^{\alpha })=\varphi (p^{\alpha })(1+\beta )+p^{\alpha -1}\cdot \delta _{\alpha \beta }=\varphi (p^{\alpha })\left( 1+\beta +\dfrac{ p^{\alpha -1}\cdot \delta _{\alpha \beta }}{\varphi (p^{\alpha })}\right) . \end{equation*} Taking into account that $\varphi (p^{\alpha })=p^{\alpha }-p^{\alpha -1}$ one obtains the first result. \newline For $\beta >\alpha $, we have $\gcd (k,p^{\alpha })\mid p^{\beta }\iff \gcd (k,p^{\alpha })\mid p^{\alpha }$. Hence, \begin{equation*} G_{p^{\beta }}(p^{\alpha })=\Sum_{\substack{ k=1 \\ \gcd (k,p^{\alpha })\mid p^{\beta }}}^{p^{\alpha }}\gcd (k,p^{\alpha })= \Sum_{\substack{ k=1 \\ \gcd (k,p^{\alpha })\mid p^{\alpha }}}^{p^{\alpha }}\gcd (k,p^{\alpha })=G_{p^{\alpha }}(p^{\alpha }) \end{equation*} and the second result follows by application of the first formula. \end{proof} \begin{remark} \label{rem2} The result of Proposition~\ref{prop-Gd-fuer-palpha} can be written in one single formula: for $p\in \mathbb{P}$, $\alpha \in \mathbb{N}$ and $\beta \in \mathbb{N}\cup \{0\}$, we have \begin{equation*} G_{p^{\beta }}(p^{\alpha })=\varphi (p^{\alpha })\left( 1+\min (\alpha ,\beta )+\dfrac{\delta _{\alpha ,\min (\alpha ,\beta )}}{p-1}\right) . \end{equation*} \end{remark} \begin{theorem} \label{Gd-als-ProduktVers1} For $n\in \mathbb{N}$ with prime powers decomposition $n=p_{1}^{\lambda _{1}}\cdot \ldots \cdot p_{t}^{\lambda _{t}}$ and positive integer $d=c\cdot p_{1}^{\kappa _{1}}\cdots p_{t}^{\kappa _{t}}$ with $p_{j}\nmid c$ for all $j=1,\dots ,t$, and $0\leqslant \kappa _{j}$ we have the representation\footnote{$\kappa _{j}=0$ means that $p_{j}$ is not present in the decomposition of $d$, i.e., $p_{j}\nmid d$.} \begin{equation*} G_{d}(n)=\varphi (n)\cdot \Prod_{j=1}^{t}\left( 1+\min (\kappa _{j},\lambda _{j})+\delta _{\lambda _{j},\min (\kappa _{j},\lambda _{j})} \dfrac{1}{p_{j}-1}\right) . \end{equation*} \end{theorem} \begin{proof} Because $G_{d}$ is multiplicative, by Theorem~\ref{Gd-mult}, and applying Lemma~\ref{lemma1}~(iii), we obtain \begin{align*} G_{d}(n)& =G_{d}\left( \Prod_{j=1}^{t}p_{j}^{\lambda _{j}}\right) \\ & =\Prod_{j=1}^{t}G_{c\cdot p_{1}^{\kappa _{1}}\cdots p_{t}^{\kappa _{t}}}\left( p_{j}^{\lambda _{j}}\right) \\ & =\Prod_{j=1}^{t}G_{p_{j}^{\kappa _{j}}}\left( p_{j}^{\lambda _{j}}\right) \\ & =\Prod_{j=1}^{t}\varphi \left( p_{j}^{\lambda _{j}}\right) \left( 1+\min (\kappa _{j},\lambda _{j})+\dfrac{\delta _{\lambda _{j}\min (\kappa _{j},\lambda _{j})}}{p_{j}-1}\right) , \end{align*} where the last equation is a consequence of Rem.\ \ref{rem2}. \end{proof} We note that under the notation of Theorem~\ref{Gd-als-ProduktVers1} the equation \begin{equation*} \gcd (d,n)=p_{1}^{\kappa _{1}}\cdots p_{t}^{\kappa _{t}} \end{equation*} defines unique numbers $\kappa _{j}$ ($j=1,\dots ,t$) with $0\leqslant \kappa _{j}\leqslant \lambda _{j}$, such that the result can be written in the form \begin{equation*} G_{d}(n)=G_{\gcd (d,n)}(n)=\varphi (n)\cdot \Prod_{j=1}^{t}\left( 1+\kappa _{j}+\delta _{\lambda _{j},\kappa _{j}}\dfrac{1}{p_{j}-1}\right) . \end{equation*} \section{Dirichlet series, averages and asymptotic properties} Some asymptotic formulas of the Broughan's gcd-sum function were derived by Broughan \cite{Broughan-2001} and Bordell\`es \cite{Bordelles-2012}. The average order of the Dirichlet series of the Broughan's gcd-sum function was studied by Broughan \cite{Broughan-2007} and Bordell\`es \cite{Bordelles-2007}. In this section we give some examples of Dirichlet series of arithmetic functions connected with $G_{d}(n)$. We calculate the average functions and derive some asymptotic formulas for these examples. The Dirichlet series for an arithmetic function $f(n)$ is defined (see, e.g., \cite[11.1, p.\ 224]{Apostol}) by \begin{equation*} \mathcal{F}(s):=\Sum_{n=1}^{\infty }\dfrac{f(n)}{n^{s}}. \end{equation*} The most prominent example is the Riemann $\zeta $ function $\zeta (s)= \Sum_{n=1}^{\infty }\dfrac{1}{n^{s}}$. It is clear, that $\zeta (s)$ is the Dirichlet series associated to $f(n)=1$, for all $n\in \mathbb{N}$. For any prime number $p$, the Bell series \cite[Sect.\ 2.15, p.\ 42ff]{Apostol} of an arithmetic function $f$ is the formal power series \begin{equation*} f_{p}(x)=\Sum_{n=0}^{\infty }f(p^{n})x^{n}. \end{equation*} If $f$ is multiplicative the corresponding Dirichlet series is given by \begin{equation*} \mathcal{F}(s)=\Sum_{n=1}^{\infty }f(n)n^{-s}= \Prod_{p}f_{p}(p^{-s}) \end{equation*} provided that the Dirichlet series converges absolutely for $\Rea s>a$ (see, e.g., \cite[Th.\ 11.7, p.\ 231]{Apostol}). The number $e\in \mathbb{N}\cup \{0\}$ is called the \emph{$m$-adic order of $n\in \mathbb{N}$} ($m\in \mathbb{N}$), if $m^{e}\mid n$ and $m^{e+1}\nmid n$. It is denoted by $e=\nu_m(n)$. \subsection{The arithmetic function \texorpdfstring{$G_d$}{}} \subsubsection{Dirichlet series} Since $G_{d}=\rho _{d}\ast \varphi $ (see (\ref{gd=rho*phi})) and \begin{align*} \mathcal{P}_{d}(s)& :=\Sum_{n=1}^{\infty }\dfrac{\rho _{d}(n)}{n^{s}}=\Sum_{n\mid d}\dfrac{1}{n^{s-1}}; \\ \Phi (s)& :=\Sum_{n=1}^{\infty }\dfrac{\varphi (n)}{n^{s}} =\dfrac{\zeta (s-1)}{\zeta (s)}=\Prod_{p}\dfrac{1-p^{-s} }{1-p^{1-s}}, \end{align*} (\cite[Ex.\ 4, p.\ 229 and p.\ 231]{Apostol}), we have according to \cite[Th. 11.5]{Apostol}: for $\Rea s>2$, \begin{equation*} \mathcal{G}_{d}(s):=\Sum_{n=1}^{\infty }\dfrac{G_{d}(n)}{n^{s}}=\Sum_{n=1}^{\infty }\dfrac{\rho _{d}(n)\ast \varphi (n)}{n^{s}}=\mathcal{P}_{d}(s)\Phi (s), \end{equation*} so \begin{equation*} \mathcal{G}_{d}(s)=\dfrac{\zeta (s-1)}{\zeta (s)}\cdot \Sum_{n\mid d}\dfrac{1}{n^{s-1}}=\Prod_{p}\dfrac{ 1-p^{-s}}{1-p^{1-s}}\cdot \Sum_{n\mid d}\dfrac{1}{n^{s-1}}. \end{equation*} If $d=1$ one obviously has $\mathcal{G}_{1}(s)=\dfrac{\zeta (s-1)}{\zeta (s)} $ (cf.\ \cite[Ex.\ 3, p.\ 231]{Apostol}). For $d\in \mathbb{P}$, one has \begin{equation*} \mathcal{G}_{d}(s)=\dfrac{\zeta (s-1)}{\zeta (s)}\left( 1+\dfrac{1}{d^{s-1}} \right) . \end{equation*} \subsubsection{Average functions} We study the asymptotic behaviour of the average function \begin{equation*} \mathcal{G}_{d}^{\left[ \alpha \right] }\left( x\right) :=\sum_{n\leq x}n^{-\alpha }G_{d}\left( n\right) \end{equation*} as $n$ tends to infinity. Taking advantage of the representation $G_{d}=\rho _{d}\ast \varphi $ we obtain \begin{equation*} \mathcal{G}_{d}^{\left[ \alpha \right] }\left( x\right) =\sum_{n\leq x}\sum_{w\mid n}\frac{\rho _{d}\left( w\right) }{w^{\alpha }}\frac{\varphi \left( \frac{n}{w}\right) }{\left( \frac{n}{w}\right) ^{\alpha }}. \end{equation*} By application of \cite[Th.\ 3.10, p.\ 65]{Apostol}, we conclude that \begin{equation*} \mathcal{G}_{d}^{\left[ \alpha \right] }\left( x\right) =\sum_{n\leq x}n^{-\alpha }\rho _{d}\left( n\right) \Phi ^{\left[ \alpha \right] }\left( \frac{x}{n}\right) =\sum_{w\mid d}w^{-\left( \alpha -1\right) }\Phi ^{\left[ \alpha \right] }\left( \frac{x}{w}\right) , \end{equation*} where $\Phi ^{\left[ \alpha \right] }$ denotes the average \begin{equation*} \Phi ^{\left[ \alpha \right] }\left( x\right) :=\sum_{n\leq x}n^{-\alpha }\varphi \left( n\right) \end{equation*} of Euler's totient function $\varphi $. We distinguish 3 cases. Because, for $\alpha \leq 1$, \begin{equation*} \Phi ^{\left[ \alpha \right] }\left( x\right) \sim \frac{x^{2-\alpha }}{ 2-\alpha }\zeta ^{-1}\left( 2\right) +O\left( x^{1-\alpha }\log x\right) \text{ \qquad }\left( x\rightarrow \infty \right) , \end{equation*} (\cite[Ex.\ 8, p.\ 71]{Apostol}), we have \begin{equation*} \mathcal{G}_{d}^{\left[ \alpha \right] }\left( x\right) \sim \frac{ x^{2-\alpha }}{2-\alpha }\zeta ^{-1}\left( 2\right) \sum_{w\mid d}\frac{1}{w} +O\left( x^{1-\alpha }\log x\right) \text{ \qquad }\left( x\rightarrow \infty \right) . \end{equation*} Because, for $\alpha >1,\alpha \neq 2$, \begin{equation*} \Phi ^{\left[ \alpha \right] }\left( x\right) \sim \frac{x^{2-\alpha }}{ 2-\alpha }\zeta ^{-1}\left( 2\right) +\frac{\zeta \left( \alpha -1\right) }{ \zeta \left( \alpha \right) }+O\left( x^{1-\alpha }\log x\right) \text{ \qquad }\left( x\rightarrow \infty \right) , \end{equation*} (\cite[Ex.\ 7, p.\ 71]{Apostol}), we have \begin{equation*} \mathcal{G}_{d}^{\left[ \alpha \right] }\left( x\right) \sim \frac{ x^{2-\alpha }}{2-\alpha }\zeta ^{-1}\left( 2\right) \sum_{w\mid d}\frac{1}{w} +\frac{\zeta \left( \alpha -1\right) }{\zeta \left( \alpha \right) } \sum_{w\mid d}w^{-\left( \alpha -1\right) }+O\left( x^{1-\alpha }\log x\right) \text{ \qquad }\left( x\rightarrow \infty \right) . \end{equation*} Finally, for $\alpha =2$, we have \begin{equation*} \Phi ^{\left[ 2\right] }\left( x\right) \sim \frac{\log x}{\zeta \left( 2\right) }+\frac{\gamma }{\zeta \left( 2\right) }-A+O\left( \frac{\log x}{x} \right) \text{ \qquad }\left( x\rightarrow \infty \right) , \end{equation*} where $\gamma $ is Euler's constant and $A=\sum_{n=1}^{\infty }\mu \left( n\right) n^{-2}\log n$ (\cite[Ex.\ 6, p.\ 71]{Apostol}), and we conclude that \begin{equation*} \mathcal{G}_{d}^{\left[ 2\right] }\left( x\right) \sim \frac{1}{\zeta \left( 2\right) }\sum_{w\mid d}\frac{\log \left( x/w\right) }{w}+\left( \frac{ \gamma }{\zeta \left( \alpha \right) }-A\right) \sum_{w\mid d}w^{-1}+O\left( \frac{\log x}{x}\right) \text{ \qquad }\left( x\rightarrow \infty \right) . \end{equation*} \subsection{The arithmetic function \texorpdfstring{$G_{n/\gcd (r,n)}(n)$}{}} Let $r\in \mathbb{N}$ be given. Consider the arithmetic function \begin{equation*} b^{(r)}(n):=G_{n/\gcd (r,n)}(n). \end{equation*} which is easily seen to be multiplicative. Let $p$ be a prime number and put $\beta =\nu_p(r)$. According to Prop.\ \ref{prop-Gd-fuer-palpha} one has \begin{equation*} b^{(r)}(p^{n})=G_{p^{n}/\gcd (r,p^{n})}(p^{n})=G_{p^{n-\beta }}(p^{n})=\varphi (p^{n})\left( 1+n-\beta +\dfrac{\delta _{n,n-\beta }}{p-1} \right) . \end{equation*} So, if $\beta =0$ one has $\delta _{n,n-\beta }=1$ and \begin{equation*} b^{(r)}(p^{n})=\varphi (p^{n})\left( 1+n+\frac{1}{p-1}\right) =(n+1)p^{n}-np^{n-1}. \end{equation*} Therefore, for $\beta =0$, the Bell series is given by \begin{equation*} b_{p}^{(r)}(x)=\Sum_{n=0}^{\infty }b^{(r)}(p^{n})x^{n}= \Sum_{n=0}^{\infty }\left( (n+1)p^{n}-np^{n-1}\right) x^{n}=\dfrac{1-x}{(1-px)^{2}}. \end{equation*} If $\beta >0$ one has $\delta _{n,n-\beta }=0$ and \begin{align*} b_{p}^{(r)}(x)& =\Sum_{n=0}^{\infty }b^{(r)}(p^{n})x^{n}= \Sum_{n=0}^{\infty }\varphi (p^{n})(1+n-\beta )x^{n} \\ & =\Sum_{n=0}^{\infty }(p^{n}-p^{n-1})(1+n-\beta )x^{n}= \dfrac{(p-1)(px\beta -\beta +1)}{p(px-1)^{2}}. \end{align*} Hence, the Dirichlet series is given by \begin{equation*} \mathcal{B}^{(r)}(s):=\dfrac{\zeta ^{2}(s-1)}{\zeta (s)}\prod\limits_{p\mid r} \dfrac{(p-1)(1-\left( 1-p^{1-s}\right) \beta \left( p\right) )}{p-p^{1-s}} \text{ \qquad }\left( \Rea s>2\right) , \end{equation*} where $\beta \left( p\right) =\nu_p(r)$. \subsection{The arithmetic function \texorpdfstring{$G_{n}(\gcd (r,n)n)$}{}} Let $r\in \mathbb{N}$ be given. Consider the arithmetic function \begin{equation*} a^{(r)}(n):=G_{n}(\gcd (r,n)\cdot n) \end{equation*} which is easily seen to be multiplicative. Let $p$ be a prime number and put $\beta =\nu_p(r)$. According to Remark \ref{rem2} one has \begin{align*} a^{(r)}(p^{n})& =G_{p^{n}}(\gcd (r,p^{n})p^{n})=G_{p^{n}}\left( p^{n+\min ( \beta ,n) }\right) \\ & =\varphi \left( p^{n+\min ( \beta ,n) }\right) \left( 1+n+\delta _{0,\min ( \beta ,n) }\frac{1}{p-1}\right) . \end{align*} If $\beta =0$, one has $\delta _{0,\min (\beta ,n)}=1$ and \begin{equation*} a^{(r)}(p^{n})=\varphi (p^{n})\left( 1+n+\frac{1}{p-1}\right) =(n+1)p^{n}-np^{n-1}. \end{equation*} Therefore, for $\beta =0$, the Bell series is given by \begin{equation*} a_{p}^{(r)}(x)=\Sum_{n=0}^{\infty }a^{(r)}(p^{n})x^{n}= \Sum_{n=0}^{\infty }\left( (n+1)p^{n}-np^{n-1}\right) x^{n}=\dfrac{1-x}{(1-px)^{2}}. \end{equation*} If $\beta >0$ one has $\delta _{0,\min (\beta ,n)}=0$ and \begin{align*} a_{p}^{(r)}(x)& =1+\Sum_{n=1}^{\infty }(1+n)\varphi (p^{n+\min (\beta ,n)})x^{n} \\ & =1+\Sum_{n=1}^{\beta }(1+n)\varphi (p^{2n})x^{n}+ \Sum_{n=\beta +1}^{\infty }(1+n)\varphi (p^{n+\beta })x^{n} \\ & =1+\Sum_{n=1}^{\beta }(1+n)(p^{2n}-p^{2n-1})x^{n}+ \Sum_{n=\beta +1}^{\infty }(1+n)(p^{n+\beta }-p^{n+\beta -1})x^{n} \\ & =1+\dfrac{p-1}{p}\Sum_{n=1}^{\beta }(n+1)(p^{2}x)^{n}+p^{\beta -1}(p-1)\Sum_{n=\beta +1}^{\infty }(n+1)(px)^{n} \\ & =1+\dfrac{(p-1)px((\beta +1)(p^{2}x)^{\beta +1}-(\beta +2)(p^{2}x)^{\beta }-p^{2}x+2)}{(p^{2}x-1)^{2}} \\ & \hspace{2cm}-\dfrac{(p-1)p^{2\beta }x^{\beta +1}((\beta +1)px-(\beta +2))}{ (px-1)^{2}}. \end{align*} Hence, the Dirichlet series is given by \begin{equation*} \mathcal{A}^{(r)}(s):=\dfrac{\zeta ^{2}(s-1)}{\zeta (s)}\prod\limits_{p\mid r}\left( \dfrac{\left( 1-p^{1-s}\right) ^{2}}{1-p^{-s}}a_{p}^{(r)}(p^{-s}) \right) \text{ \qquad }\left( \Rea s>2\right) . \end{equation*} where \begin{eqnarray*} a_{p}^{(r)}(p^{-s}) &=&1-\dfrac{(p-1)p^{2\beta }p^{-s(\beta +1)}((\beta +1)p^{1-s}-(\beta +2))}{(1-p^{1-s})^{2}} \\ &&+\dfrac{(p-1)p^{1-s}((\beta +1)p^{(2-s)(\beta +1)}-(\beta +2)p^{(2-s)\beta }-p^{2}x+2)}{(p^{2}x-1)^{2}} \end{eqnarray*} and $\beta =\beta \left( p\right) =\nu_p(r)$. \section{Acknowledgment} The authors are grateful to the referee for valuable remarks, in particular for pointing out the similarity of our function $H_n(r)$ to the famous Erd\H{o}s multiplication table problem. \begin{thebibliography}{99} \bibitem{Apostol} T. 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Illinois, 2010, \\ \url{http://dms.umontreal.ca/~koukoulo/documents/publications/phdthesis.pdf}. \bibitem{Kou2013-2} D. Koukoulopolous, On the number of integers in a generalized multiplication table, preprint, \url{http://arxiv.org/abs/1102.3236v3}. \bibitem{Tenenbaum} G. Tenenbaum, Sur la probabilit\'e qu'un entier poss\`ede un diviseur dans un intervalle donn\'e, {\em Compositio Math.} {\bf51} (1984), 243--263. \end{thebibliography} \bigskip \hrule \bigskip \noindent 2010 {\it Mathematics Subject Classification}: Primary 11A05; Secondary 11A25, 11F66, 11N37, 11N56. \noindent \emph{Keywords: } multiplicative structure, arithmetic function, one-variable Dirichlet series, asymptotic results on arithmetic functions. \bigskip \hrule \bigskip \vspace*{+.1in} \noindent Received February 6 2013; revised version received June 29 2013. Published in {\it Journal of Integer Sequences}, July 29 2013. \bigskip \hrule \bigskip \noindent Return to \htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}. \vskip .1in \end{document}