<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\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 $\varepsilon &gt;0.$ 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 $\varepsilon &gt;0,$\ 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 $\varepsilon &gt;0,$\
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)  &amp;=&amp;\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\}  \\
&amp;=&amp;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) 
\\
&amp;=&amp;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 $\func{Re}s&gt;1,$ 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 $\func{Re}s&gt;1,$ 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}

                                                                                

</pre></body></html>