\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}}} \def\func#1{\mathop{\rm #1}\nolimits}% \begin{document} \begin{center} \epsfxsize=4in \leavevmode\epsffile{logo129.eps} \end{center} \begin{center} \vskip 1cm{\LARGE\bf A Note on the Average Order of \\ \vskip .1in the gcd-sum Function } \vskip 1cm \large Olivier Bordell\`{e}s\\ 2, All\'{e}e de la Combe\\ La Boriette\\ 43000 Aiguilhe\\ France\\ \href{mailto:borde43@wanadoo.fr}{\tt borde43@wanadoo.fr}\\ \end{center} \vskip .2in \begin{abstract} We prove an asymptotic formula for the average order of the gcd-sum function by using a new convolution identity. \end{abstract} \vskip .2in \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}{Proposition}[section] \newtheorem{corollary}{Corollary}[section] \newtheorem{lemma}{Lemma}[section] %\input{tcilatex} \theoremstyle{definition} \theoremstyle{remark} \numberwithin{equation}{section} \newcommand{\thmref}[1]{Theorem~\ref{#1}} \newcommand{\secref}[1]{\S\ref{#1}} \newcommand{\lemref}[1]{Lemma~\ref{#1}} % \input tcilatex \section{Introduction and main result} In 2001, Broughan \cite{Broughan} studied the gcd-sum function $g$ defined for any positive integer $n$ by \begin{equation*} g\left( n\right) =\sum_{k=1}^n\left( k,n\right) , \end{equation*} where $\left( a,b\right) $ denotes the greatest common divisor of $a$ and $% b. $ The author showed that $g$ is multiplicative, and satisfies the convolution identity \begin{equation} g=\varphi *\text{ Id,} \tag{1} \end{equation} where $\varphi $ is the Euler totient function, Id is the completely multiplicative function defined by Id$\left( n\right) =n$ and $*$ is the usual Dirichlet convolution product$.$ The function $g$ appears in a specific lattice point problem \cite{Broughan,Loveless}, where it can be used to estimate the number of integer coordinate points under the square-root curve. As a multiplicative function, the question of its average order naturally arises. By using the Dirichlet hyperbola principle, Broughan \cite[Theorem 4.7]{Broughan} proved the following result: for any real number $x\geqslant 1,$ the following estimate \begin{equation} \sum_{n\leqslant x}g\left( n\right) =\frac{x^2\log x}{2\zeta \left( 2\right) }+\frac{\zeta \left( 2\right) ^2}{2\zeta \left( 3\right) }x^2+O\left( x^{3/2}\log x\right) \tag{2} \end{equation} holds. The aim of this paper is to prove another convolution identity for $g,$ and then get a fairly more precise estimate than $\left( 2\right) .$ In what follows, $\tau $ is the well-known divisor function, $\mu $ is the M% \"{o}bius function, $\mathbf{1}$ is the completely multiplicative function defined by $\mathbf{1}\left( n\right) =1,$ $F*G$ is the Dirichlet convolution product of the arithmetical functions $F$ and $G$, and we denote by $\theta $ the smallest positive real number such that \begin{equation} \sum_{n\leqslant x}\tau \left( n\right) =x\log x+x\left( 2\gamma -1\right) +O_{_\varepsilon }\left( x^{\theta +\varepsilon }\right) \tag{3} \end{equation} holds for any real numbers $x\geqslant 1$ and $\varepsilon >0.$ The following inequality \begin{equation*} \theta \geqslant \frac 14 \end{equation*} is well-known \cite{Hardy}. On the other hand, Huxley \cite{Huxley} showed that \begin{equation*} \theta \leqslant \frac{131}{416}\approx 0.3149\ldots \end{equation*} holds. Now we are able to prove the following result \begin{theorem} For any real numbers $x\geqslant 1$ and $\varepsilon >0,$\ we have \begin{equation*} \sum_{n\leqslant x}g\left( n\right) =\frac{x^2\log x}{2\zeta \left( 2\right) }+\frac{x^2}{2\zeta \left( 2\right) }\left( \gamma -\frac 12+\log \left( \frac{\mathcal{A}^{12}}{2\pi }\right) \right) +O_{_\varepsilon }\left( x^{1+\theta +\varepsilon }\right) \end{equation*} where $\mathcal{A}\approx 1.282\;427\;129\ldots $\ is the Glaisher-Kinkelin constant. \end{theorem} For further details about the Glaisher-Kinkelin constant, see \cite{Finch,Kinkelin}$.$ The reader interested in gcd-sum integer sequences should refer to Sloane's sequence \seqnum{A018804}. \section{A convolution identity} The proof uses the following lemmas. \begin{lemma} \vspace{0.5cm}For any real number $z\geqslant 1$\ and any $\varepsilon >0,$\ we have \begin{equation*} \sum_{n\leqslant z}n\tau \left( n\right) =\frac{z^2}2\log z+z^2\left( \gamma -\frac 14\right) +O_{_\varepsilon }\left( z^{1+\theta +\varepsilon }\right) . \end{equation*} \end{lemma} %TCIMACRO{\TeXButton{Proof}{\proof}} %BeginExpansion \proof% %EndExpansion The result follows easily from $\left( 3\right) $ and Abel's summation.% %TCIMACRO{\TeXButton{End Proof}{\endproof}} %BeginExpansion \endproof% %EndExpansion \begin{lemma} We have \begin{equation*} g=\mu *\left( \text{Id}\cdot \tau \right) . \end{equation*} \end{lemma} %TCIMACRO{\TeXButton{Proof}{\proof}} %BeginExpansion \proof% %EndExpansion Since $\varphi =$ $\mu *\,$Id, we have, using $\left( 1\right),$% \begin{equation*} g=\varphi *\text{Id}=\mu *\,\left( \text{Id}*\text{Id}\right) =\mu *\left( \text{Id}\cdot \tau \right) \end{equation*} which is the desired result.% %TCIMACRO{\TeXButton{End Proof}{\endproof}} %BeginExpansion \endproof% %EndExpansion \section{Proof of Theorem 1.1} By using Lemma 2.2, we get \begin{equation*} \sum_{n\leqslant x}g\left( n\right) =\sum_{d\leqslant x}\mu \left( d\right) \sum_{k\leqslant x/d}k\tau \left( k\right) \end{equation*} and Lemma 2.1 applied to the inner sum gives \begin{eqnarray*} \sum_{n\leqslant x}g\left( n\right) &=&\sum_{d\leqslant x}\mu \left( d\right) \left\{ \frac{x^2}{d^2}\left( \frac 12\log \left( \frac xd\right) +\gamma -\frac 14\right) +O_{_\varepsilon }\left( \left( \frac xd\right) ^{1+\theta +\varepsilon }\right) \right\} \\ &=&x^2\left\{ \left( \frac 12\log x+\gamma -\frac 14\right) \sum_{d\leqslant x}\frac{\mu \left( d\right) }{d^2}-\sum_{d\leqslant x}\frac{\mu \left( d\right) \log d}{2d^2}\right\} +O_{_\varepsilon }\left( x^{1+\theta +\varepsilon }\sum_{d\leqslant x}\frac 1{d^{1+\theta +\varepsilon }}\right) \\ &=&x^2\left\{ \left( \frac 12\log x+\gamma -\frac 14\right) \sum_{d=1}^\infty \frac{\mu \left( d\right) }{d^2}-\sum_{d=1}^\infty \frac{% \mu \left( d\right) \log d}{2d^2}+O\left( \frac{\log x}x\right) \right\} +O_{_\varepsilon }\left( x^{1+\theta +\varepsilon }\right) . \end{eqnarray*} Now it is well-known that, for $s\in \mathbb{C}$ such that $\func{Re}s>1,$ we have \begin{equation*} \frac 1{\zeta \left( s\right) }=\sum_{d=1}^\infty \frac{\mu \left( d\right) }{d^s} \end{equation*} which gives by differentiation \begin{equation*} \frac{\zeta ^{\,\prime }\left( s\right) }{\left( \zeta \left( s\right) \right) ^2}=\sum_{d=1}^\infty \frac{\mu \left( d\right) \log d}{d^s} \end{equation*} for $\func{Re}s>1,$ and hence \begin{equation*} \sum_{n\leqslant x}g\left( n\right) =\frac{x^2}{2\zeta \left( 2\right) }% \left( \log x-\frac{\zeta ^{\,\prime }\left( 2\right) }{\zeta \left( 2\right) }+2\gamma -\frac 12\right) +O_{_\varepsilon }\left( x^{1+\theta +\varepsilon }\right) , \end{equation*} and we use \begin{equation*} \frac{\zeta ^{\,\prime }\left( 2\right) }{\zeta \left( 2\right) }=\gamma -\log \left( \frac{\mathcal{A}^{12}}{2\pi }\right) . \end{equation*} The proof of the theorem is now complete.% %TCIMACRO{\TeXButton{End Proof}{\endproof}} %BeginExpansion \endproof% %EndExpansion \textbf{Acknowledgment}. I am grateful to the referee for his valuable suggestions on this paper. \begin{thebibliography}{1} \bibitem{Broughan} K. A. Broughan, \href{http://www.cs.uwaterloo.ca/journals/JIS/VOL4/BROUGHAN/gcdsum.html}{The gcd-sum function}, {\it J. Integer Sequences} \textbf{4} (2001), Art. 01.2.2. \bibitem{Finch} S. R. Finch, \textit{Mathematical Constants}, Cambridge University Press, 2003, pp.\ 135--145. \bibitem{Hardy} G. H. Hardy, The average order of the arithmetical functions $P\left( x\right) $ and $\Delta \left( x\right) $, {\it Proc. London Math. Soc.} \textbf{15} $\left( 2\right) $ (1916), 192--213. \bibitem{Huxley} M. N. Huxley, Exponential sums and lattice points III, {\it Proc. London Math. Soc.} \textbf{87} (2003), 591--609. \bibitem{Kinkelin} H. Kinkelin, \"{U}ber eine mit der Gammafunktion verwandte Transcendente und deren Anwendung auf die Integralrechnung, {\it J. Reine Angew. Math.\ } \textbf{57} (1860), 122--158. \bibitem{Loveless} A. D. Loveless, The general GCD-product function, {\it Integers} \textbf{6} (2006), article A19, available at \href{http://www.integers-ejcnt.org/vol6.html}{\tt http://www.integers-ejcnt.org/vol6.html}. Corrigendum: {\bf 6} (2006), article A39. \end{thebibliography} \bigskip \hrule \bigskip \noindent 2000 {\it Mathematics Subject Classification}: Primary 11A25; Secondary 11N37. \noindent \emph{Keywords: } gcd-sum function, Dirichlet convolution, average order of multiplicative functions . \bigskip \hrule \bigskip \noindent (Concerned with sequence \seqnum{A018804}.) \bigskip \hrule \bigskip \vspace*{+.1in} \noindent Received July 10 2006; revised version received March 28 2007. Published in {\it Journal of Integer Sequences}, March 28 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}