\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} \setlength{\textwidth}{6.5in} \setlength{\oddsidemargin}{.1in} \setlength{\evensidemargin}{.1in} \setlength{\topmargin}{-.5in} \setlength{\textheight}{8.9in} \newcommand{\seqnum}[1]{\href{http://oeis.org/#1}{\underline{#1}}} \begin{document} \begin{center} \epsfxsize=4in \leavevmode\epsffile{logo129.eps} \end{center} \begin{center} \vskip 1cm{\LARGE\bf Mean Values of Generalized gcd-sum \\ \vskip .12in and lcm-sum Functions } \vskip 1cm \large Olivier Bordell\`es \\ 2, All\' ee de la Combe\\ La Boriette\\ 43000 Aiguilhe\\ France\\ \href{mailto:borde43@wanadoo.fr}{\tt borde43@wanadoo.fr}\\ \end{center} \vskip .2 in \begin{abstract} We consider a generalization of the gcd-sum function, and obtain its average order with a quasi-optimal error term. We also study the reciprocals of the gcd-sum and lcm-sum functions. \end{abstract} \newtheorem{theorem}{Theorem} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{def1}{Definition}[section] \newtheorem{def2}[def1]{Definition} \newtheorem{def3}{Definition}[section] \newtheorem{theo1}[def1]{Theorem} \newtheorem{theo2}{Theorem}[section] \newtheorem{theo3}[def3]{Theorem} \newtheorem{theo4}{Theorem}[section] \newtheorem{lem1}{Lemma}[section] \newtheorem{lem2}{Lemma}[section] \newtheorem{lem3}[def3]{Lemma} \newtheorem{lem4}[theo4]{Lemma} \section{Introduction and notation} The so-called gcd-sum function, defined by $$g\left( n\right) =\sum_{j=1}^n\left( n,j\right)$$ where $\left( a,b\right) $ denotes the greatest common divisor of $a$ and $b,$ was first introduced by Broughan (\cite{brou,brou2}) who studied its main properties, and showed among other things that $g$ satisfies the convolution identity (see also the beginning of the proof of Lemma~\ref{lm1}) $$g=\varphi \ast \text{Id}$$ where $F \ast G$ is the usual Dirichlet convolution product. By using the following alternative convolution identity $$g=\mu \ast \left( \text{Id}\cdot \tau \right) ,$$ where $\mu $ is the M\"obius function and $\tau $ is the divisor function, we were able in \cite{bor1} to get the average order of $g.$ Our result can be stated as follow. If $\theta $ is the exponent in the Dirichlet divisor problem, then the following asymptotic formula \begin{equation} \begin{split} \sum_{n\leqslant x}g\left( n\right) =\dfrac{x^2\log x}{2\zeta \left( 2\right) }+\dfrac{x^2}{2\zeta \left( 2\right) }\left( \gamma -\dfrac 12+\log \left( \dfrac{\mathcal{A}^{12}}{2\pi }\right) \right) +O_{\varepsilon }\left( x^{1+\theta +\varepsilon }\right) \end{split} \end{equation} holds for any real number $\varepsilon >0,$ where $\mathcal{A}\approx 1.282\;427\;129\ldots $\ is the Glaisher-Kinkelin constant. The inequality $\theta \geqslant 1/4$ is well-known, and, from the work of Huxley \cite{hux} we know that $\theta \leqslant 131/416\approx 0.3149.$\\\\ The aim of this paper is first to work with a function generalizing the function $g$ and prove an asymptotic formula for its average order similarly as in $\left( 1\right) .$ In sections 5, 6 and 7 we will establish estimates for the lcm-sum function, and for reciprocals of the gcd-sum and lcm-sum functions. We begin with classical notation. \bigskip \noindent 1. {\it Multiplicative functions}. The following arithmetic functions are well-known. \begin{equation*} \begin{split} \text{Id}^a\left( n\right) &=n^a\quad \left( a\in \mathbb{Z}^{*}\right)\\ {\bf 1}\left( n\right) &=1\\ \end{split} \end{equation*} and $\mu$, $\varphi$, $\sigma_k$ and $\tau_k$ are respectively the M\"obius function, the Euler totient function, the sum of $k$th powers of divisors function and the $k$th Piltz divisor function. Recall that $\tau _k$ can be defined by $\tau _k=\underset{k \, \mathrm{times}}{\underbrace{\mathbf{1} \ast \cdots \ast {\mathbf{1}}}}$ for any integer $k\geqslant 1$ and that $\tau _2=\tau .$ We also have $\sigma _k=\sum_{d\mid n}d^k$ and $\sigma _0=\tau .$ \bigskip \noindent 2. {\it Exponent in the Dirichlet-Piltz divisor problem}. For any integer $k\geqslant 2,$ $\theta _k$ is defined to be the smallest positive real number such that the asymptotic formula \begin{equation} \begin{split} \sum_{n\leqslant x}\tau _k\left( n\right) =x\mathcal{P}_{k-1}\left( \log x\right) +O_{\varepsilon ,k}\left( x^{\theta _k+\varepsilon }\right) \end{split} \end{equation} holds for any real number $\varepsilon >0.$ Here $\mathcal{P}_{k-1}$ is a polynomial of degree $k-1$ with real coefficients, the leading coefficient being $\frac{1}{\left( k-1\right) !}.$ It is now well-known that $\frac 13\leqslant \theta _3\leqslant \frac{43}{96}$ and that $\frac{k-1}{2k}\leqslant \theta _k\leqslant \frac{k-1}{k+2}$ for $k\geqslant 4$ (see \cite{ivi}, for example).\\\\ By convention, we set $$\tau _0\left( n\right) = \begin{cases} 1, & \mathrm{if\ } n=1;\\ 0, & \mathrm{otherwise;}\end{cases}$$ and $\theta _1=0$. \section{A generalization of the gcd-sum function} \begin{def1} We define the sequence of arithmetic functions $f_{k,j}\left( n\right) $ in the following way.\\\\ $\left( i\right) $ For any integers $j,n\geqslant 1$, we set\\\\ \begin{equation*} \begin{split} f_{1,j}\left( n\right) &= \begin{cases} 1, & \mathrm{if\ } (n,j)=1;\\ 0, & \mathrm{otherwise;} \end{cases}\\ f_{2,j}\left( n\right) &= \begin{cases} (n,j), & \mathrm{if\ } j \leqslant n;\\ 0, & \mathrm{otherwise}. \end{cases} \end{split} \end{equation*} $\left( ii\right) $ For any integers $j\geqslant 1$ and $k\geqslant 3,$ we set $$f_{k,j}=f_{2,j}*\left( \text{Id}\cdot \tau _{k-2}\right) .$$ \end{def1} \begin{def2} For any integers $n,k\geqslant 1,$ we define the sequence of arithmetic functions $g_k\left( n\right)$ by $$g_k\left( n\right) =\sum_{j=1}^nf_{k,j}\left( n\right) .$$ \end{def2} \paragraph{Examples.} \begin{equation*} \begin{split} g_1\left( n\right) &=\underset{\left( n,j\right) =1}{\sum_{j=1}^n}1=\varphi \left( n\right),\\ g_2\left( n\right) &=\sum_{j=1}^n\left( j,n\right) =g\left( n\right),\\ g_3\left( n\right) &=\sum_{j=1}^n\underset{d\geqslant j}{\sum_{d\mid n}}\frac nd\left( j,d\right),\\ &\vdots\\ g_k\left( n\right) &=\sum_{j=1}^n\sum_{d_{k-2}\mid n}\sum_{d_{k-3}\mid d_{k-2}}\cdots \underset{d_1\geqslant j}{\sum_{d_1\mid d_2}}\frac n{d_1}\left( j,d_1\right). \end{split} \end{equation*} Now we are able to state the following result. \begin{theo1} \label{th1} Let $\varepsilon >0$ be any real number and $k\geqslant 1$ any integer. Then, for any real number $x\geqslant 1$ sufficiently large, we have $$\sum_{n\leqslant x}g_k\left( n\right) =\frac{x^2}{2\zeta \left( 2\right) }\mathcal{R}_{k-1}\left( \log x\right) +O_{\varepsilon ,k}\left( x^{1+\theta _k+\varepsilon }\right)$$ where $\mathcal{R}_{k-1}$\ is a polynomial of degree $k-1$ and leading coefficient $\frac 1{\left( k-1\right) !}.$ The following table gives $\mathcal{R}_{k-1}$\ for $k\in \left\{ 1,2,3\right\}$ \begin{center} \begin{tabular}{|c|c|c|c|} \hline k & $\mathrm{1}$ & $\mathrm{2}$ & $\mathrm{3}$\\ \hline & & &\\ $\mathcal{R}_{k-1}$ & $\mathrm{1}$ & $X+\gamma -\dfrac 12+\log \left( \dfrac{\mathcal{A}^{12}}{2\pi }\right) $ & $\dfrac{X^2}2+\alpha X+\beta $\\ & & & \\ \hline \end{tabular} \end{center} where \begin{equation*} \begin{split} \alpha &=2\gamma -\dfrac 12+\log \left( \dfrac{\mathcal{A}^{12}}{2\pi }\right)\\ \beta &=-\frac{\zeta ^{\,\prime \prime }\left( 2\right) }{2\zeta \left( 2\right) }+\left( \gamma -\log \left( \dfrac{\mathcal{A}^{12}}{2\pi }\right) \right) ^2-\left( 3\gamma -\frac 12\right) \left( \gamma -\log \left( \dfrac{\mathcal{A}^{12}}{2\pi }\right) \right)\\ &-\frac 14\left( 12\gamma _1-12\gamma ^2+6\gamma -1\right) \end{split} \end{equation*} and \begin{center} \begin{tabular}{|c|c|} \hline $\mathrm{Constant}$ & $\mathrm{Name}$\\ \hline $\gamma \approx 0.577\;215\;664\ldots $ & $\mathrm{Euler-Mascheroni}$\\ \hline $\gamma _1\approx -0.072\;815\;845\ldots $ & $\mathrm{Stieltjes}$\\ \hline $\mathcal{A}\approx 1.282\;427\;129\ldots $ & $\mathrm{Glaisher-Kinkelin}$\\ \hline \end{tabular} \end{center} \end{theo1} \section{Main properties of the function $g_k$} The following lemma lists the main tools used in the proof of Theorem~\ref{th1}. \begin{lem1} \label{lm1} For any integer $k\geqslant 1,$\ we have $$g_{k+1}=g_k \ast \mathrm{Id}$$ and then $$g_k=\varphi \ast \left( \mathrm{Id}\cdot \tau _{k-1}\right) .$$ Moreover, we have \begin{equation} \begin{split} g_k=\mu \ast \left( \mathrm{Id}\cdot \tau _k\right) . \end{split} \end{equation} Thus, the Dirichlet series $G_k\left( s\right) $\ of $g_k$\ is absolutely convergent in the half-plane $\Re s>2,$\ and has an analytic continuation to a meromorphic function defined on the whole complex plane with value $$G_k\left( s\right) =\frac{\zeta \left( s-1\right) ^k}{\zeta \left( s\right) }.$$ \end{lem1} \begin{proof} Broughan already proved the first relation for $k=1$ (see \cite[Thm.\ 4.7]{brou}), but, for the sake of completeness, we give here another proof. \begin{equation*} \begin{split} \left( g_1*\text{Id}\right) \left( n\right) &=\left( \varphi *\text{Id}\right) \left( n\right) =\sum_{d\mid n}d\varphi \left( \frac nd\right)\\ &=\sum_{d\mid n}d\underset{\left( k,n/d\right) =1}{\sum_{k\leqslant n/d}}1=\sum_{d\mid n}d\underset{\left( j,n\right) =d}{\overset{n}{\sum_{j=1}}}1\\ &=\sum_{j=1}^n\left( j,n\right) =\sum_{j=1}^nf_{2,j}\left( n\right) =g_2\left( n\right). \end{split} \end{equation*} For $k=2,$ we get $$\left( g_2*\text{Id}\right) \left( n\right) =\sum_{d\mid n}\frac nd\sum_{j=1}^d\left( j,d\right) =\sum_{j=1}^n\underset{d\geqslant j}{\sum_{d\mid n}}\frac nd\left( j,d\right) =\sum_{j=1}^nf_{3,j}\left( n\right) =g_3\left( n\right) .$$ Now let us suppose $k \geqslant 3.$ We have \begin{equation*} \begin{split} g_{k+1}\left( n\right) &=\sum_{j=1}^nf_{k+1,j}\left( n\right) =\sum_{j=1}^n\left( f_{2,j}*\left( \text{Id}\cdot \tau _{k-1}\right) \right) \left( n\right)\\ &=\sum_{j=1}^n\left( f_{2,j}*\text{Id}\cdot \tau _{k-2}*\text{Id}\right) \left( n\right)\\ &=\sum_{d\mid n}\frac nd\sum_{j=1}^d\left( f_{2,j}*(\text{Id}\cdot \tau _{k-2})\right) \left( d\right)=\left( g_k*\text{Id}\right) \left( n\right). \end{split} \end{equation*} The second relation is easily shown by induction. For the third, we have using $\varphi =\mu \ast \mathrm{Id} $ \begin{equation*} \begin{split} g_k&=\varphi *\,\left( \text{Id}\cdot \tau _{k-1}\right) =\mu *\left( \text{Id}*\,(\text{Id}\cdot \tau _{k-1})\right)\\ &=\mu *\left( \text{Id}\cdot \left( {\bf 1}*\tau _{k-1}\right) \right) =\mu *\left( \text{Id}\cdot \tau _k\right). \end{split} \end{equation*} The last proposition comes from the equality $\left( 3\right)$ $$ g_k=\mu *\left( \text{Id}\cdot \tau _k\right) =\mu *\,\underset{k \, \mathrm{ times}}{\underbrace{\text{Id}*\cdots *\text{Id}}}$$ and the Dirichlet series of $\mu $ and Id. \end{proof} \section{Proof of Theorem 1} \begin{lem2} \label{lm2} For any integer $k\geqslant 1$ and any real numbers $x>1$ and $\varepsilon >0,$\ we have $$\sum_{n\leqslant x}n\tau _k\left( n\right) =x^2 \mathcal{Q}_{k-1}\left( \log x\right) +O_{\varepsilon ,k}\left( x^{1+\theta _k+\varepsilon }\right)$$ where $\mathcal{Q}_{k-1}$\ is a polynomial of degree $k-1$ and leading coefficient $\frac 1{2\left( k-1\right) !}.$ \end{lem2} \begin{proof} Using summation by parts and $\left( 2\right) $, we get \begin{equation*} \begin{split} \sum_{n\leqslant x}n\tau _k\left( n\right) &=x\sum_{n\leqslant x}\tau _k\left( n\right) -\int_1^x\left( \sum_{n\leqslant t}\tau _k\left( n\right) \right) dt\\ &=x^2\mathcal{P}_{k-1}\left( \log x\right) +O_{\varepsilon ,k}\left( x^{1+\theta _k+\varepsilon }\right) -\int_1^x\left( t\mathcal{P}_{k-1}\left( \log t\right) +O_{\varepsilon ,k}\left( t^{\theta _k+\varepsilon }\right) \right) dt. \end{split} \end{equation*} Writing $$\mathcal{P}_{k-1}\left( X\right) =\sum_{j=0}^{k-1}a_jX^j$$ with $a_{k-1}=\frac 1{\left( k-1\right) !},$ we obtain $$\sum_{n\leqslant x}n\tau _k\left( n\right) =x^2\sum_{j=0}^{k-1}a_j\left( \log x\right) ^j-\sum_{j=0}^{k-1}a_j\int_1^xt\left( \log t\right) ^jdt+O_{\varepsilon ,k}\left( x^{1+\theta _k+\varepsilon }\right)$$ and the formula $$\int_1^xt\left( \log t\right) ^jdt=x^2\sum_{i=0}^j\left( -1\right) ^{j-i}\frac{j!}{2^{j+1-i}\times i!}\left( \log x\right) ^i-\left( -1\right) ^j\frac{j!}{2^{j+1}}$$ (easily proved by induction) gives \begin{equation*} \begin{split} \sum_{n\leqslant x}n\tau _k\left( n\right) &=x^2\sum_{j=0}^{k-1}a_j\left\{ \left( \log x\right) ^j-\sum_{i=0}^j\left( -1\right) ^{j-i}\frac{j!}{2^{j+1-i}\times i!}\left( \log x\right) ^i\right\}\\ &+\sum_{i=0}^j\left( -1\right) ^j\frac{j!\,a_j}{2^{j+1}}+O_{\varepsilon ,k}\left( x^{1+\theta _k+\varepsilon }\right)\\ &=x^2\sum_{j=0}^{k-1}a_j\left\{ \frac{\left( \log x\right) }2^j-\sum_{i=0}^{j-1}\left( -1\right) ^{j-i}\frac{j!}{2^{j+1-i}\times i!}\left( \log x\right) ^i\right\} +O_{\varepsilon ,k}\left( x^{1+\theta _k+\varepsilon }\right) \end{split} \end{equation*} which completes the proof of the lemma. \end{proof} \paragraph{Remark.} For $k=3,$ the following result is well-known (see \cite[Exer. II.3.4]{ten}, for example) $$\sum_{n\leqslant x}\tau _3\left( n\right) =x\left\{ \frac{\left( \log x\right) ^2}2+\left( 3\gamma -1\right) \log x+3\gamma ^2-3\gamma -3\gamma _1+1\right\} +O_\varepsilon \left( x^{\theta _3+\varepsilon }\right)$$ and gives \begin{equation} \begin{split} \sum_{n\leqslant x}n\tau _3\left( n\right) =x^2\left\{ \frac{\left( \log x\right) ^2}4+\left( \frac{6\gamma -1}4\right) \log x-\frac{12\gamma _1-12\gamma ^2+6\gamma -1}8\right\} +O_\varepsilon \left( x^{1+\theta _3+\varepsilon }\right). \end{split} \end{equation} \vspace{0.5cm} Now we are able to prove Theorem~\ref{th1}. Using $\left( 3\right) $ we get $$\sum_{n\leqslant x}g_k\left( n\right) =\sum_{d\leqslant x}\mu \left( d\right) \sum_{m\leqslant x/d}m\tau _k\left( m\right) ,$$ and lemma~\ref{lm1} gives \begin{equation*} \begin{split} \sum_{n\leqslant x}g_k\left( n\right) &=\sum_{d\leqslant x}\mu \left( d\right) \left\{ \left( \frac xd\right) ^2\mathcal{Q}_{k-1}\left( \log \frac xd\right) +O_{\varepsilon ,k}\left( \left( \frac xd\right) ^{1+\theta _k+\varepsilon }\right) \right\}\\ &=x^2\sum_{d\leqslant x}\frac{\mu \left( d\right) }{d^2}\mathcal{Q}_{k-1}\left( \log \frac xd\right) +O_{\varepsilon ,k}\left( x^{1+\theta _k+\varepsilon }\right). \end{split} \end{equation*} Writing $$\mathcal{Q}_{k-1}\left( X\right) =\sum_{j=0}^{k-1}b_jX^j$$ with $b_{k-1}=\frac 1{2\left( k-1\right) !},$ we get \begin{equation*} \begin{split} \sum_{n\leqslant x}g_k\left( n\right) &=x^2\sum_{d\leqslant x}\frac{\mu \left( d\right) }{d^2}\sum_{j=0}^{k-1}b_j\left( \log \frac xd\right) ^j+O_{\varepsilon ,k}\left( x^{1+\theta _k+\varepsilon }\right)\\ &=x^2\sum_{j=0}^{k-1}\sum_{h=0}^j\binom jhb_j\left( \log x\right) ^{j-h}\sum_{d\leqslant x}\left( -1\right) ^h\frac{\mu \left( d\right) }{d^2}\left( \log d\right) ^h+O_{\varepsilon ,k}\left( x^{1+\theta _k+\varepsilon }\right) \end{split} \end{equation*} and the equality \begin{equation*} \begin{split} \sum_{d\leqslant x}\left( -1\right) ^h\frac{\mu \left( d\right) }{d^2}\left( \log d\right) ^h &=\sum_{d=1}^{\infty}\left( -1\right) ^h\frac{\mu \left( d\right) }{d^2}\left( \log d\right) ^h-\sum_{d>x}\left( -1\right) ^h\frac{\mu \left( d\right) }{d^2}\left( \log d\right) ^h\\ &=\left[ \frac{d^h}{ds^h}\left( \frac 1{\zeta \left( s\right) }\right) \right] _{\left[ s=2\right] }+O\left( \frac{\left( \log x\right) ^h}x\right), \end{split} \end{equation*} implies \begin{equation*} \begin{split} \sum_{n\leqslant x}g_k\left( n\right) &=x^2\sum_{j=0}^{k-1}\sum_{h=0}^j\binom jh\left( \left[ \frac{d^h}{ds^h}\left( \frac 1{\zeta \left( s\right) }\right) \right] _{\left[ s=2\right] }\right) b_j\left( \log x\right) ^{j-h}\\ &+O\left( x\left( \log x\right) ^{k-1}\right) +O_{\varepsilon ,k}\left( x^{1+\theta _k+\varepsilon }\right)\\ &=x^2\sum_{j=0}^{k-1}\sum_{h=0}^j\binom jh\left( \left[ \frac{d^h}{ds^h}\left( \frac 1{\zeta \left( s\right) }\right) \right] _{\left[ s=2\right] }\right) b_j\left( \log x\right) ^{j-h}+O_{\varepsilon ,k}\left( x^{1+\theta _k+\varepsilon }\right), \end{split} \end{equation*} and writing $$\left[ \frac{d^h}{ds^h}\left( \frac 1{\zeta \left( s\right) }\right) \right] _{\left[ s=2\right] }=\frac{A_h}{2\zeta \left( 2\right) ^{h+1}}$$ with $A_h\in \mathbb{R}$ (and $A_0=2$), we obtain $$\sum_{n\leqslant x}g_k\left( n\right) =\frac{x^2}{2\zeta \left( 2\right) }\sum_{j=0}^{k-1}\sum_{h=0}^j\binom jh\frac{A_hb_j\left( \log x\right) ^{j-h}}{\zeta \left( 2\right) ^h}+O_{\varepsilon ,k}\left( x^{1+\theta _k+\varepsilon }\right)$$ which is the desired result. The leading coefficient is $\binom{k-1}0A_0b_{k-1}=\frac 1{\left( k-1\right) !}.$ The particular cases are easy to check.\\\\ (i) For $k=1,$ the result is well-known (see \cite[Exer.\ 4.14]{bor2}) $$\sum_{n\leqslant x}g_1\left( n\right) =\sum_{n\leqslant x}\varphi \left( n\right) =\frac{x^2}{2\zeta \left( 2\right) }+O\left( x\log x\right) .$$\\\\ (ii) For $k=2,$ see \cite{bor1}.\\\\ (iii) For $k=3,$ we use $\left( 4\right) $ and the computations made above. The proof of Theorem~\ref{th1} is now complete. \section{Sums of reciprocals of the gcd} The purpose of this section is to prove the following estimate. \begin{theo2} \label{th2} For any real number $x > e$ sufficiently large,\ we have $$\sum_{n\leqslant x}\left( \sum_{j=1}^n\frac 1{\left( j,n\right) }\right) =\frac{\zeta \left( 3\right) }{2\zeta \left( 2\right) }\,x^2+O\left( x\left( \log x\right) ^{2/3}\left( \log \log x\right) ^{4/3}\right) .$$ \end{theo2} \begin{proof} For any integer $n\geqslant 1,$ we set $$\mathcal{G}\left( n\right) =\sum_{j=1}^n\frac 1{\left( j,n\right) }.$$ With a similar argument used in the proof of the identity $g=\varphi \ast \mathrm{Id}$ (see lemma~\ref{lm1}), it is easy to check that $$\mathcal{G}=\varphi \ast \mathrm{Id}^{-1}\text{,}$$ and thus $$\sum_{n\leqslant x}\mathcal{G}\left( n\right) =\sum_{d\leqslant x}\frac 1d\sum_{m\leqslant x/d}\varphi \left( m\right) .$$ The well-known result (see \cite{wal}, for example) $$\sum_{n\leqslant x}\varphi \left( n\right) =\frac{x^2}{2\zeta \left( 2\right) }+O\left( x\left( \log x\right) ^{2/3}\left( \log \log x\right) ^{4/3}\right) ,$$ combined with some classical computations, allows us to conclude the proof of Theorem~\ref{th2}. \end{proof} \section{The lcm-sum function} \begin{def3} For any integer $n\geqslant 1,$ we define $$l\left( n\right) =\sum_{j=1}^n\left[ n,j\right]$$ where $\left[ a,b\right] $ is the least common multiple of $a$ and $b$. \end{def3} \begin{lem3} \label{lm3} We have the following convolution identity $$l=\frac 12\left( (\mathrm{Id}^2\cdot \left( \varphi +\tau _0\right)) *\mathrm{Id}\right) .$$ \end{lem3} \begin{proof} We have $$\sum_{j=1}^n\frac j{\left( n,j\right) }=\sum_{d\mid n}\frac 1d\underset{\left( n,j\right) =d}{\sum_{j=1}^n}j=\sum_{d\mid n}\frac 1d\underset{\left( k,n/d\right) =1}{\sum_{k\leqslant n/d}}kd=\sum_{d\mid n}\underset{\left( k,n/d\right) =1}{\sum_{k\leqslant n/d}}k,$$ with \begin{equation*} \begin{split} \underset{\left( k,N\right) =1}{\sum_{k\leqslant N}}k &=\sum_{d\mid N}d\mu \left( d\right) \sum_{m\leqslant N/d}m\\ &=\frac 12\sum_{d\mid N}d\mu \left( d\right) \left\{ \frac Nd\left( \frac Nd+1\right) \right\}\\ &=\frac N2\sum_{d\mid N}\mu \left( d\right) \left( \frac Nd+1\right) =\frac N2\left( \varphi +\tau _0\right) \left( N\right), \end{split} \end{equation*} and hence $$\sum_{j=1}^n\frac j{\left( n,j\right) }=\frac 12\sum_{d\mid n}\frac nd\left( \varphi +\tau _0\right) \left( \frac nd\right) =\frac 12\left(( \text{Id}\cdot \left( \varphi +\tau _0\right)) * {\bf 1}\right) \left( n\right) ,$$ and we conclude by noting that $$l\left( n\right) =n\sum_{j=1}^n\frac j{\left( n,j\right)}$$ which completes the proof, since Id is completely multiplicative. \end{proof} \begin{theo3} \label{th3} For any real number $x > e$ sufficiently large,\ we have the following estimate $$\sum_{n\leqslant x}\left( \sum_{j=1}^n\left[ n,j\right] \right) =\frac{\zeta \left( 3\right) }{8\zeta \left( 2\right) }\,x^4+O\left( x^3\left( \log x\right) ^{2/3}\left( \log \log x\right) ^{4/3}\right) .$$ \end{theo3} \begin{proof} Using lemma~\ref{lm3}, we get \begin{equation*} \begin{split} \sum_{n\leqslant x}l\left( n\right) &=\frac 12\sum_{d\leqslant x}d\sum_{m\leqslant x/d}m^2\left( \varphi +\tau _0\right) \left( m\right)\\ &=\frac 12\sum_{d\leqslant x}d\sum_{m\leqslant x/d}m^2\varphi \left( m\right) +O\left( x^2\right) \end{split} \end{equation*} and the estimation (see \cite{wal}) $$\sum_{n\leqslant x}n^2\varphi \left( n\right) =\frac{x^4}{4\zeta \left( 2\right) }+O\left( x^3\left( \log x\right) ^{2/3}\left( \log \log x\right) ^{4/3}\right)$$ implies \begin{equation*} \begin{split} \sum_{n\leqslant x}l\left( n\right) &=\frac 12\sum_{d\leqslant x}d\left\{ \frac 1{4\zeta \left( 2\right) }\left( \frac xd\right) ^4+O\left( \left( \frac xd\right) ^3\left( \log x\right) ^{2/3}\left( \log \log x\right) ^{4/3}\right) \right\} +O\left( x^2\right)\\ &=\frac{x^4}{8\zeta \left( 2\right) }\sum_{d=1}^\infty \frac 1{d^3} + O \left ( x^3 \left( \log x \right) ^{2/3} \left( \log \log x \right)^{4/3} \right ) +O \left ( x^2 \right ), \end{split} \end{equation*} which is the desired result. \end{proof} \section{Sum of reciprocals of the lcm} We will prove the following result. \begin{theo4} \label{th4} For any real number $x>1$ sufficiently large,\ we have $$\sum_{n\leqslant x}\left( \sum_{j=1}^n\frac 1{\left[ n,j\right] }\right) =\frac{\left( \log x\right) ^3}{6\zeta \left( 2\right) }+\frac{\left( \log x\right) ^2}{2\zeta \left( 2\right) }\left( \gamma +\log \left( \frac{\mathcal{A}^{12}}{2\pi }\right) \right) +O\left( \log x\right) .$$ \end{theo4} \vspace{0.5cm} Some useful estimates are needed. \begin{lem4} \label{lm4} Set $C_{\varphi }=\log \left( \dfrac{\mathcal{A}^{12}}{2\pi }\right) \approx 1.147\;176\ldots $ For any real number $x\geqslant 1,$\ we have \begin{equation*} \begin{split} (i) & : \sum_{n\leqslant x}\frac{\varphi \left( n\right) }{n^2}=\frac{\log x}{\zeta \left( 2\right) }+\frac{C_{\varphi }}{\zeta \left( 2\right) }+O\left( \frac{\log ex}x\right).\\ (ii) & : \sum_{n\leqslant x}\frac{\varphi \left( n\right) }{n^2}\log \left( \frac xn\right) =\frac{\left( \log x\right) ^2}{2\zeta \left( 2\right) }+\frac{C_{\varphi }\log x}{\zeta \left( 2\right) }+O\left( 1\right).\\ (iii) & : \frac 12\sum_{n\leqslant x}\frac{\varphi \left( n\right) }{n^2}\left( \log \left( \frac xn\right) \right) ^2=\frac{\left( \log x\right) ^3}{6\zeta \left( 2\right) }+\frac{C_{\varphi }\left( \log x\right) ^2}{2\zeta \left( 2\right) }+O\left( \log x\right). \end{split} \end{equation*} \end{lem4} \begin{proof} $\left( i\right) .$ Using $\varphi =\mu *$ Id, we get \begin{equation*} \begin{split} \sum_{n\leqslant x}\frac{\varphi \left( n\right) }{n^2} &= \sum_{d\leqslant x}\frac{\mu \left( d\right) }{d^2}\sum_{m\leqslant x/d}\frac 1m\\ &= \sum_{d\leqslant x}\frac{\mu \left( d\right) }{d^2}\left\{ \log \left( \frac xd\right) +\gamma +O\left( \frac dx\right) \right\}\\ &= \left( \log x+\gamma \right) \sum_{d\leqslant x}\frac{\mu \left( d\right) }{d^2}-\sum_{d\leqslant x}\frac{\mu \left( d\right) \log d}{d^2}+O\left( \frac 1x\sum_{d\leqslant x}\frac 1d\right)\\ &=\frac{\log x}{\zeta \left( 2\right) }+\frac \gamma {\zeta \left( 2\right) }-\frac{\zeta ^{\,\prime }\left( 2\right) }{\left( \zeta \left( 2\right) \right) ^2}+O\left( \frac{\log ex}x\right). \end{split} \end{equation*} Recall that $\dfrac{\zeta ^{\,\prime }\left( 2\right) }{\zeta \left( 2\right) }=\gamma -C_{\varphi }$.\\\\ $\left( ii\right) $ and $\left( iii\right) .$ Abel's summation and estimate $\left( i \right) .$ We leave the details to the reader. \end{proof} \vspace{0.5cm} Now we are able to show Theorem~\ref{th4}. For any integer $n\geqslant 1,$ we set $$\mathcal{L}\left( n\right) =\sum_{j=1}^n\frac 1{\left[ n,j\right] }.$$ Since \begin{equation*} \begin{split} \mathcal{L}\left( n\right) &= \frac 1n\sum_{j=1}^n\frac{\left( n,j\right) }j=\frac 1n\sum_{d\mid n}d\underset{\left( j,n\right) =d}{\sum_{j=1}^n}\frac 1j\\ &= \frac 1n\sum_{d\mid n}d\underset{\left( k,n/d\right) =1}{\sum_{k\leqslant n/d}}\frac 1{kd}=\frac 1n\sum_{d\mid n}\underset{\left( k,n/d\right) =1}{\sum_{k\leqslant n/d}}\frac 1k, \end{split} \end{equation*} we get \begin{equation*} \begin{split} \sum_{n\leqslant x}\mathcal{L}\left( n\right) &=\sum_{n\leqslant x}\frac 1n\sum_{d\mid n}\underset{\left( k,n/d\right) =1}{\sum_{k\leqslant n/d}}\frac 1k\\ &=\sum_{d\leqslant x}\frac 1d\sum_{h\leqslant x/d}\frac 1h\underset{\left( k,h\right) =1}{\sum_{k\leqslant h}}\frac 1k\\ &=\sum_{d\leqslant x}\frac 1d\sum_{h\leqslant x/d}\frac 1h\sum_{\delta \mid h}\frac{\mu \left( \delta \right) }\delta \sum_{m\leqslant h/\delta }\frac 1m\\ &=\sum_{d\leqslant x}\frac 1d\sum_{\delta \leqslant x/d}\frac{\mu \left( \delta \right) }{\delta ^2}\sum_{a\leqslant x/\left( d\delta \right) }\frac 1a\sum_{m\leqslant a}\frac 1m\\ &=\sum_{d\leqslant x}\sum_{\delta d\leqslant x}\frac 1d\frac{\mu \left( \delta \right) }{\delta ^2}\sum_{a\leqslant x/\left( d\delta \right) }\frac 1a\sum_{m\leqslant a}\frac 1m\\ &=\sum_{n\leqslant x}\frac 1{n^2}\sum_{d\mid n}d\mu \left( \frac nd\right) \sum_{a\leqslant x/n}\frac 1a\sum_{m\leqslant a}\frac 1m, \end{split} \end{equation*} and the convolution identity $\varphi =\mu \ast \mathrm{Id}$ implies that $$\sum_{n\leqslant x}\mathcal{L}\left( n\right) =\sum_{n\leqslant x}\frac{\varphi \left( n\right) }{n^2}\sum_{a\leqslant x/n}\frac 1a\sum_{m\leqslant a}\frac 1m.$$ Thus \begin{equation*} \begin{split} \sum_{n\leqslant x}\mathcal{L}\left( n\right) &=\sum_{n\leqslant x}\frac{\varphi \left( n\right) }{n^2}\sum_{a\leqslant x/n}\frac 1a\left\{ \log a+\gamma +O\left( \frac 1a\right) \right\}\\ &=\sum_{n\leqslant x}\frac{\varphi \left( n\right) }{n^2}\left\{ \frac 12\left( \log \frac xn\right) ^2+\gamma \left( \log \frac xn\right) +O\left( 1\right) \right\}\\ &=\frac{\left( \log x\right) ^3}{6\zeta \left( 2\right) }+\frac{C_{\varphi }\left( \log x\right) ^2}{2\zeta \left( 2\right) }+\frac{\gamma \left( \log x\right) ^2}{2\zeta \left( 2\right) }+O\left( \log x\right) \end{split} \end{equation*} where $C_{\varphi }=\log \left( \dfrac{\mathcal{A}^{12}}{2\pi }\right) ,$ which concludes the proof.\\\\ \section{Acknowledgment} I am indebted to the referee for his valuable suggestions. \begin{thebibliography}{9} \bibitem{bor1} O. Bordell\`es, \href{http://www.cs.uwaterloo.ca/journals/JIS/VOL10/Bordelles/bordelles90.html}{A note on the average order of the gcd-sum function}, {\it J. Integer Sequences} {\bf 10} (2007), Article 07.3.3. \bibitem{bor2} O. Bordell\`es, {\it Th\`emes d'Arithm\'etique}, Editions Ellipses, 2006. \bibitem{brou} K. A. Broughan, \href{http://www.cs.uwaterloo.ca/journals/JIS/VOL4/BROUGHAN/gcdsum.html}{\tt The gcd-sum function}, {\it J. Integer Sequences} {\bf 4} (2001), Article 01.2.2. \\ Errata, April 16, 2007, \href{http://www.cs.uwaterloo.ca/journals/JIS/VOL4/BROUGHAN/errata1.pdf}{\tt http://www.cs.uwaterloo.ca/journals/JIS/VOL4/BROUGHAN/errata1.pdf}. \bibitem{brou2} K. A. Broughan, \href{http://www.cs.uwaterloo.ca/journals/JIS/VOL10/Broughan/broughan1.html}{The average order of the Dirichlet series of the gcd-sum function}, {\it J. Integer Sequences} {\bf 10} (2007), Article 07.4.2. \bibitem{hux} M. N. Huxley, Exponential sums and lattice points III, {\it Proc. London Math. Soc.} {\bf 87} (2003), 591--609. \bibitem{ivi} A. Ivi\'c, E. Kr\"atzel, M. K\"uhleitner, and W. G. Nowak, Lattice points in large regions and related arithmetic functions: Recent developments in a very classic topic, in W. Schwarz and J. Steuding, eds., {\it Conference on Elementary and Analytic Number Theory}, Frank Steiner Verlag, 2006, pp.\ 89--128. \bibitem{ten} G. Tenenbaum, {\it Introduction \`a la Th\'eorie Analytique et Probabiliste des Nombres}, Soci\'et\'e Math\'ematique de France, 1995. \bibitem{wal} A. Walfisz, {\it Weylsche Exponentialsummen in der neueren Zahlentheorie}, Leipzig BG Teubner, 1963. \end{thebibliography} \bigskip \hrule \bigskip \noindent 2000 {\it Mathematics Subject Classification}: Primary 11A25; Secondary 11N37. \noindent \emph{Keywords: } greatest common divisor, average order, Dirichlet convolution. \bigskip \hrule \bigskip \vspace*{+.1in} \noindent Received July 2 2007; revised version received August 27 2007. Published in {\it Journal of Integer Sequences}, August 27 2007. \bigskip \hrule \bigskip \noindent Return to \htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}. \vskip .1in \end{document}