\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}}} \DeclareMathOperator*{\Res}{Res} \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 Mean Values of a Class of \\ \vskip .1in Arithmetical Functions} \vskip 1cm \large Deyu Zhang\footnote{This work is supported by National Natural Science Foundation of China(Grant Nos.\ 10771127, 11001154) and Shandong Province Natural Science Foundation (Nos.\ BS2009SF018, ZR2010AQ009).}\\ School of Mathematical Sciences\\ Shandong Normal University\\ Jinan 250014\\ Shandong\\ P. R. China\\ \href{mailto:zdy_78@yahoo.com.cn}{\tt zdy\_78@yahoo.com.cn}\\ \ \\ Wenguang Zhai$^1$\\ Department of Mathematics\\ China University of Mining and Technology \\ Beijing, 100083\\ P. R. China\\ \href{mailto:zhaiwg@hotmail.com}{\tt zhaiwg@hotmail.com} \\ \end{center} \vskip .2in \begin{abstract} In this paper we consider a class of functions $\mathcal {U}$ of arithmetical functions which include $\tilde{P}(n)/n$, where $\tilde{P}(n):=n \prod_{p|n}(2-\frac{1}{p})$. For any given $U\in\mathcal {U}$, we obtain the asymptotic formula for $\sum_{n\leq x}U(n)$, which improves a result of De Koninck and K\'atai. \end{abstract} \newcommand{\bvec}[1]{\mbox{\boldmath$#1$}} \renewcommand{\d}{\displaystyle} \renewcommand\thefootnote{\fnsymbol{footnote}} \section{Introduction} In 1933, Pillai \cite{s8} introduced the function $$P(n)=\sum_{k=1}^{n}\gcd(k,n),$$ and proved that $$P(n)=\sum_{d\mid n}d\varphi(n/d), \ \ \ \ \text{and} \ \ \ \ \sum_{d\mid n}P(d)=n d(n)=\sum_{d\mid n}\sigma(d)\varphi(n/d),$$ where $\varphi$ is Euler's function, $d(n)$ and $\sigma(n)$ denote the number of divisors of $n$ and the sum of the divisors of $n$ respectively. Many authors investigated the properties of $P(n)$, see $\cite{s1, s2, s3, s4, s5, s8, s11};$ it is Sloane's sequence \seqnum{A018804}. Chidambaraswamy and Sitaramachandrarao \cite{s5} showed that, given an arbitrary $\epsilon >0$, $$\sum_{n\leq x}P(n)=e_1x^2\log x+e_2x^2+O(x^{1+\theta+\epsilon}),$$ where $e_1, e_2$ are computable constants and $0<\theta<1/2$ is some exponent contained in \begin{eqnarray}\label{eq:E1} \sum_{n\leq x}d(n)=x\log x+(2\gamma-1)x+O(x^{\theta+\epsilon}).\end{eqnarray} The asymptotic formula \eqref{eq:E1} is the well-known Dirichlet divisor problem. The latest value of $\theta$ is $\theta=131/416$ proved by Huxley \cite{H}. T\'oth \cite{s10} first defined the gcd-sum function over regular integers modulo $n$ by the relation \begin{eqnarray}\label{eq:E2} \tilde{P}(n)=\sum_{k\in {\rm Reg}_n } \gcd(k,n), \end{eqnarray} where ${\rm Reg}_n=\{k:1\leq k\leq n \ \text{and }k \ \ \text{is regular (mod $n$})\}$, and proved that $\tilde{P}(n)$ is multiplicative and for every $n\geq 1,$ \begin{eqnarray}\label{eq:E3} \tilde{P}(n)=n \prod_{p|n}(2-\frac{1}{p}). \end{eqnarray} It is sequence \seqnum{A176345} in Sloane's Encyclopedia. He also obtained the following asymptotic formula \begin{eqnarray}\label{eq:E4} \sum_{n\leq x}\tilde{P}(n)=\frac{x^{2}}{2\zeta(2)}(K_1 \log x+K_2)+O(x^{3/2}\delta(x)), \end{eqnarray} where $K_1$ and $K_2$ are certain constants and $\delta(x)$ is given by $$\delta(x)=\exp(-A(\log x)^{3/5}(\log\log x)^{-1/5}).$$ Zhang and Zhai \cite{s13} showed that the estimate of $\sum_{n\leq x}\tilde{P}(n)$ is closely related to the square-free divisor problem and improved the error term of \eqref{eq:E4} under RH. De Koninck and K\'atai \cite{s6} introduced two wide classes of arithmetical functions $\mathcal {R}$ and $\mathcal {U}$, the first of which includes the function $P(n)/n$, and the second of which includes $\tilde{P}(n)/n$. More precisely, the class $\mathcal {R}$ is made of the following functions $R$. Firstly let $\gamma(n)$ denote the kernel of $n\geq 2,$ that is $\gamma(n)=\prod_{p\mid n}p$ (with $\gamma(1)=1$). Then, given an arbitrary positive constant $c$, an arbitrary real number $\alpha >0$ and a multiplicative function $\kappa (n)$ satisfying $\mid \kappa(n)\mid\leq\frac{c}{\gamma(n)^{\alpha}}$ for all $n\geq 2,$ let $R\in \mathcal {R}$ be defined by \begin{eqnarray}\label{eq:E5} R(n)=R_{\kappa,c,\alpha}(n):=d(n)\sum_{d\parallel n}\kappa (d)=d(n)\prod_{p^{a}\parallel n}(1+\kappa (p^{a})). \end{eqnarray} It is easily seen that if we let $\kappa(p^{a})=-\frac{a/(a+1)}{p},$ then the corresponding function $R(n)$ is precisely $P(n)/n$. De Koninck and K\'atai \cite{s6} showed that \begin{eqnarray}\label{eq:E6} T(x):=\sum_{n\leq x}R(n)=A_0 x\log x+B_0 x+O(x^{\beta+\epsilon}), \end{eqnarray} with $$\beta=\begin{cases} \theta, &\text{if}\ \ \alpha\geq 1-\theta; \\ 1-\alpha, &\text{if}\ \ \alpha< 1-\theta; \end{cases}$$ where $\theta$ is the exponent in \eqref{eq:E1}, $A_0, B_0$ are certain constants. As for the class of functions $\mathcal {U}$, it is made of the functions $$U(n)=U_{h,c,\alpha}(n):=2^{\omega(n)}\sum_{d\mid n}h(d),$$ where $\omega(n)$ stands for the number of distinct prime factors of $n$, and $h$ is a multiplicative function satisfying $|h(n)|\leq\frac{c}{\gamma(n)^{\alpha}}$ for all $n\geq 2.$ It is easily seen that by taking $h(p)=-\frac{1}{2p}$ and $h(p^{a})=0,$ for $a\geq 2,$ we obtain the particular case $U(n)=\tilde{P}(n)/n$. De Koninck and K\'atai \cite{s6} proved that \begin{eqnarray}\label{eq:E7} S(x):=\sum_{n\leq x}U(n)=t_1 x\log x+t_2 x+O(\frac{x}{\log x}), \end{eqnarray} where $t_1, t_2$ are certain constants. In this paper, we shall prove the following \begin{theorem} \label{thm:1} Suppose $0\leq \alpha<1.$ Then we have \begin{eqnarray}\label{eq:E8} S(x)=t_1 x\log x+t_2 x+O(x^{1-\alpha+\epsilon}+x^{1/2+\epsilon}). \end{eqnarray} \end{theorem} \begin{remark} (i) From our proof we see that the evaluation of $S(x)$ is closely related to the distribution of the zeros of the Riemann zeta function. The exponent $1/2$ can be reduced to $4/11$ if RH is true. (ii) The exponent $1-\alpha $ in the error term of Theorem \ref{thm:1} is best possible when $\alpha $ is small. For example, if we take $h(n)=n^{-\alpha}$ with $0<\alpha<1/2$, then our proof with slight modifications yields $$\sum_{n\leq x}U(n)=t_1 x\log x+t_2 x+t_3 x^{1-\alpha}\log x+t_4 x^{1-\alpha}+O(x^{1/2+\epsilon}).$$\end{remark} We are also interested in the short interval case. In this case, the restrictions on $\alpha $ and RH can be removed. Actually, we have the following Theorem \ref{thm:3}. \begin{theorem} \label{thm:3} Suppose \eqref{eq:E1} holds for $1/4<\theta <1/3$. Then for $x^{\theta+2\epsilon}\leq y\leq x,$ we have \begin{eqnarray}\label{eq:E9} \sum_{x1$. Then $$ \sum_{n=1}^{\infty}\frac{U(n)}{n^{s}}=\frac{\zeta^{2}(s)}{\zeta(2s)}G(s),$$ where $G(s)$ can be written as a Dirichlet series $G(s)= \sum\limits_{n=1}^{\infty}\frac{g(n)}{n^{s}}$, which is absolutely convergent for $\Re s>1-\alpha$. Moreover $g(n)$ satisfies $|g(n)|\ll n^{-\alpha+\epsilon}$. \end{lemma} \begin{proof} For $\Re s>1,$ by Euler product representation we have $$F(s):=\sum_{n=1}^{\infty}\frac{U(n)}{n^{s}}=\prod_{p}\left(1+\sum_{\beta=1}^{\infty}\frac{U(p^{\beta})}{p^{\beta s}}\right),$$ where $U(p^{\beta})=2(1+h(p)+\cdots+h(p^{\beta})), \beta \geq 1.$ Thus $$\begin{array}{lll} &\displaystyle 1+\sum_{\beta=1}^{\infty}\frac{U(p^{\beta})}{p^{\beta s}}&\displaystyle =1+\sum_{\beta=1}^{\infty}\frac{2}{p^{\beta s}}+2\sum_{\beta=1}^{\infty}p^{-\beta s}\sum_{j=1}^{\beta}h(p^{j})\\[12pt] &\displaystyle &\displaystyle=\frac{1-p^{-2s}}{(1-p^{-s})^{2}}+2\sum_{\beta=1}^{\infty}p^{-\beta s}\sum_{j=1}^{\beta}h(p^{j})\\[12pt] &\displaystyle &\displaystyle=\frac{1-p^{-2s}}{(1-p^{-s})^{2}}\times\left(1+\frac{2(1-p^{-s})^{2}}{1-p^{-2s}}\sum_{\beta=1}^{\infty}p^{-\beta s}\sum_{j=1}^{\beta}h(p^{j})\right), \end{array}$$ hence we get $$ \sum_{n=1}^{\infty}\frac{U(n)}{n^{s}}=\frac{\zeta^{2}(s)}{\zeta(2s)}G(s),$$ where $$G(s)=\prod_{p}\left(1+\frac{2(1-p^{-s})^{2}}{1-p^{-2s}}\sum_{\beta=1}^{\infty}p^{-\beta s}\sum_{j=1}^{\beta}h(p^{j})\right).$$ From the above formula, it is easy to see that $G(s)$ can be expanded to a Dirichlet series $G(s)=\sum\limits_{n=1}^{\infty}\frac{g(n)}{n^{s}}$, which is absolutely convergent for $\Re s>1-\alpha$, if we notice that $|h(p)|\leq \frac{c}{p^{\alpha}}$. Therefore $|g(n)|\ll n^{-\alpha+\epsilon}$. \end{proof} \begin{lemma}\label{lem:5} Let $$\sum_{n=1}^{\infty}\frac{d^{(2)}(n)}{n^{s}}=\frac{\zeta^{2}(s)}{\zeta(2s)}, \ \ \ \Re s>1, $$ where $d^{(2)}(n)$ denote the number of square-free divisors of $n$. Then for any real numbers $x\geq 1,$ we have $$ D^{(2)}(x):=\sum_{n\leq x}d^{(2)}(n)=c_1x\log x+c_2x+\Delta^{(2)}(x)$$ with $\Delta^{(2)}(x)=O(x^{1/2}\log x)$, where $$c_1=\frac{1}{\zeta(2)},\ \ \ c_2=\frac{2\gamma-1}{\zeta(2)}-\frac{2\zeta'(2)}{\zeta^{2}(2)}.$$ \end{lemma} Moreover, if RH is true, then $\Delta^{(2)}(x)=O(x^{4/11+\epsilon} ).$ \begin{proof} The first result is due to Mertens \cite{s7} and the second one is due to Baker \cite{s0}. \end{proof} \begin{lemma}\label{lem:6} $$ \sum_{n\leq x}|g(n)|\ll x^{1-\alpha+\epsilon}.$$ \end{lemma} \begin{proof} It follows from $|g(n)|\ll n^{-\alpha+\epsilon}$. \end{proof} \begin{lemma}\label{lem:7} Let $k\geq 2$ be a fixed integer , $1x^\epsilon}}1.$$ Then we have $$ {\cal A}(x,y;k,\epsilon)\ll yx^{-\epsilon}+x^{1/4}. $$ \end{lemma} \begin{proof} This is Lemma 3 of Zhai \cite{s12}. \end{proof} \section{\bf Proof of Theorem \ref{thm:1} } Notice that \begin{eqnarray}\label{eq:E10}\frac{\zeta^{2}(s)}{\zeta(2s)}= \sum\limits_{\ell=1}^{\infty}\frac{d^{(2)}(\ell)}{\ell^{s}}, \ \ \ G(s)=\sum\limits_{m=1}^{\infty}\frac{g(m)}{m^{s}}. \end{eqnarray} By the Dirichlet convolution, we have $$\sum_{n\leq x}U(n)=\sum_{m\ell\leq x}g(m)d^{(2)}(\ell)=\sum_{m\leq x}g(m)\sum_{\ell\leq x/m}d^{(2)}(l),$$ and Lemma \ref{lem:5} applied to the inner sum gives $$ \sum_{n\leq x}U(n)=\sum_{m\leq x}g(m)\left\{\frac{c_1x}{m}\log(\frac{x}{m})+\frac{c_2x}{m}+O\left((\frac{x}{m})^{1/2+\epsilon}\right)\right\} $$ $$ =c_1 x\left\{\left(\log x+\frac{c_2}{c_1}\right)\sum_{m\leq x}\frac{g(m)}{m}-\sum_{m\leq x}\frac{g(m)\log m}{m}\right\}+O\left(x^{1/2+\epsilon}\sum_{m\leq x}\frac{|g(m)|}{m^{1/2+\epsilon}}\right)$$ $$ =c_1 x\left\{\left(\log x+\frac{c_2}{c_1}\right)\sum\limits_{m=1}^{\infty}\frac{g(m)}{m}-\sum\limits_{m=1}^{\infty}\frac{g(m)\log m}{m}+O(x^{-\alpha+\epsilon})\right\}+O\left(x^{1/2+\epsilon}\sum_{m\leq x}\frac{|g(m)|}{m^{1/2+\epsilon}}\right),$$ if we notice by Lemma \ref{lem:6} that both of the infinite series $\sum_{m=1}^{\infty}\frac{g(m)}{m}, \ \ \ \sum_{m=1}^{\infty}\frac{g(m)\log m}{m} $ are absolutely convergent, and \begin{eqnarray}\label{eq:E11}\sum_{m> x}\frac{g(m)}{m}\ll x^{-\alpha+\epsilon}, \ \ \ \ \sum_{m> x}\frac{g(m)\log m}{m}\ll x^{-\alpha+\epsilon}.\end{eqnarray} Then we have \begin{eqnarray}\label{eq:E12} \sum_{n\leq x}U(n)=t_1 x\log x+t_2 x+O(x^{1-\alpha+\epsilon})+O\left(x^{1/2+\epsilon}\sum_{m\leq x}\frac{|g(m)|}{m^{1/2+\epsilon}}\right),\end{eqnarray} where $$t_1 =\frac{1}{\zeta(2)}\sum\limits_{m=1}^{\infty}\frac{g(m)}{m}=\frac{G(1)}{\zeta(2)},\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $$ $$\ \ \ \ \ \ \ \ \ t_2=\frac{1}{\zeta(2)}\left\{(2\gamma-1-\frac{2\zeta'(2)}{\zeta(2)})\sum\limits_{m=1}^{\infty}\frac{g(m)}{m} -\sum\limits_{m=1}^{\infty}\frac{g(m)\log m}{m}\right\}$$ $$ =\frac{1}{\zeta(2)}\left\{(2\gamma-1-\frac{2\zeta'(2)}{\zeta(2)})G(1) -G'(1)\right\}. \ \ \ \ \ \ \ \ $$ By Lemma \ref{lem:6}, we have $$\sum_{m\leq x}\frac{|g(m)|}{m^{1/2+\epsilon}}\leq \sum_{m\leq x}\frac{1}{m^{1/2+\alpha+\epsilon}}\leq \begin{cases} x^{\epsilon}, \ \ \ &\alpha\geq 1/2; \\ x^{1/2-\alpha+\epsilon}, \ \ \ &\alpha<1/2, \end{cases}$$ Theorem \ref{thm:1} follows from the above estimates and Eq.~\eqref{eq:E12}. \section{\bf Proof of Theorem \ref{thm:3} } By Lemma \ref{lem:4}, we have $$ U(n)=\sum_{n=n_1n_2n_3^{2}}d(n_1)g(n_2)\mu(n_3),$$ where $d(n)$ is the divisor function. Then \begin{eqnarray}\label{eq:E13} \sum_{x x^{\epsilon}}}d(n_1)|g(n_2)|,\ \ \ \ \ \ \ \ $$ $$\Sigma_3=\sum_{{x x^{\epsilon}}}d(n_1)|g(n_2)|.\ \ \ \ \ \ \ \ $$ Recalling \eqref{eq:E1}, the inner sum in $\Sigma_1$ is $$ \frac{(x+y)}{n_2n_3^{2}}\log\frac{(x+y)}{n_2n_3^{2}}- \frac{x}{n_2n_3^{2}}\log\frac{x}{n_2n_3^{2}}+(2\gamma-1)\frac{y}{n_2n_3^{2}} +O\left(\frac{x^{\theta}}{n_2^{\theta}n_3^{2\theta}}\right)\ \ \ \ \ \ \ \ \ \ $$ $$ \ \ =\frac{(x+y)\log(x+y)-x\log x}{n_2n_3^{2}}- y\frac{\log (n_2n_3^{2})}{n_2n_3^{2}}+(2\gamma-1)\frac{y}{n_2n_3^{2}} +O\left(\frac{x^{\theta}}{n_2^{\theta}n_3^{2\theta}}\right).$$ Inserting the above expression into $\Sigma_1$ and after some easy calculations, we get \begin{eqnarray}\label{eq:E14} \Sigma_1=H(x+y)-H(x)+O\left(yx^{-\epsilon}+y^{-\alpha\epsilon+\epsilon^{2}}+x^{\theta+\epsilon}\right). \end{eqnarray} For $\Sigma_2$, we have $$|g(n_2)|\ll n_2^{-\alpha+\epsilon}\ll x^{-\alpha\epsilon+\epsilon^{2}},$$ if we notice that $n_2>x^{\epsilon}$, and hence $$\Sigma_2\ll x^{-\alpha\epsilon+\epsilon^{2}}\sum_{x x^{\epsilon}}}1 \ll x^{\epsilon^{2}}\sum_{{x x^{\epsilon}}}d(n)$$ $$ \ll x^{2\epsilon^{2}}\sum_{{x x^{\epsilon}}}1 =x^{2\epsilon^{2}}{\cal A}(x,y;2,\epsilon)$$ \begin{eqnarray}\label{eq:E16}\ll yx^{-\epsilon+2\epsilon^{2}}+x^{1/4+\epsilon^{2}}. \ \ \ \ \ \ \ \ \ \ \ \ \ \end{eqnarray} Then Theorem \ref{thm:3} follows from Eqs.~\eqref{eq:E13}--\eqref{eq:E16}. \section{Acknowledgments} The authors express their gratitude to the referee for a careful reading of the manuscript and many valuable suggestions, which highly improve the quality of this paper. \begin{thebibliography}{99} \bibitem{s0} R. C. Baker, The square-free divisor problem II, \textit{Quart. J. Math. Oxford Ser. (2)} \textbf{47} (1996), 133--146. \bibitem{s1} O. Bordell\`es, A note on the average order of the gcd-sum function, \textit{J. Integer Sequences} \textbf{10} (2007), \href{http://www.cs.uwaterloo.ca/journals/JIS/VOL10/Bordelles/bordelles90.html} {Article 07.3.3}. \bibitem{s2} O. Bordell\`es, Mean values of generalized gcd-sum and lcm-sum functions, \textit{J. Integer Sequences} \textbf{10} (2007), \href{http://www.cs.uwaterloo.ca/journals/JIS/VOL10/Bordelles2/bordelles61.html} {Article 07.9.2}. \bibitem{s3} K. Broughan, The gcd-sum function, \textit{J. Integer Sequences} \textbf{4} (2001), \href{http://www.cs.uwaterloo.ca/journals/JIS/VOL4/BROUGHAN/gcdsum.html} {Article 01.2.2}. \bibitem{s4}K. Broughan, The average order of the Dirichlet series of the gcd-sum function, \textit{J. Integer Sequences} \textbf{10} (2007), \href{http://www.cs.uwaterloo.ca/journals/JIS/VOL10/Broughan/broughan1.html}{Article 07.4.2}. \bibitem{s5} J. Chidambaraswamy and R. Sitaramachandrarao, Asymptotic results for a class of arithmetical functions, \textit{Monatsh. Math.} \textbf{99} (1985), 19--27. \bibitem{s6} J. M. de Koninck and I. K\'atai, Some remarks on a paper of L.Toth, \textit{J. Integer Sequences} \textbf{13} (2010), \href{http://www.cs.uwaterloo.ca/journals/JIS/VOL13/DeKoninck/dekoninck7.html}{Article 10.1.2}. \bibitem{H} M. N. Huxley, Exponential sums and Lattice points III, \textit{Proc. London Math. Soc.,} \textbf{87} (3) (2003), 591--609. \bibitem{s7}F. Mertens, \"Uber einige asymptotische Gesetze der Zahlentheorie, \textit{J. Reine Angew. Math.} \textbf{77} (1874), 289--338. \bibitem{s8} S. S. Pillai, On an arithmetic function, \textit{ J. Annamalai Univ.} \textbf{2} (1933), 243--248. \bibitem{s9} H. E. Richert, \"Uber die Anzahl Abelscher Gruppen gegebener Ordnung I, \textit{Math. Z.} {\bf 56} (1952), 21--32; II. ibid. \textbf{58} (1953), 71--84. \bibitem{s10} L. T\'oth, A gcd-sum function over regular integers modulo $n$, \textit{J. Integer Sequences} \textbf{12} (2009), \href{http://www.cs.uwaterloo.ca/journals/JIS/VOL12/Toth/toth3.html}{Article 09.2.5}. \bibitem{s11} Y. Tanigawa and W. Zhai, On the gcd-sum function, \textit{J. Integer Sequences} \textbf{11} (2008), \href{http://www.cs.uwaterloo.ca/journals/JIS/VOL11/Tanigawa/tanigawa12.html}{Article 08.2.3}. \bibitem{s12} W. G. Zhai, Square-free numbers as sums of two squares, in {\it Number Theory}, {Developments in Mathematics,} Springer, \textbf{15} (2006), 219--227. \bibitem{s13} D. Zhang and W. Zhai, Mean values of a gcd-sum function over regular integers modulo n, \textit{J. Integer Sequences} \textbf{13} (2010), \href{http://www.cs.uwaterloo.ca/journals/JIS/VOL13/Zhang/zhang10.html}{Article 10.4.7}. \end{thebibliography} \bigskip \hrule \bigskip \noindent 2010 {\it Mathematics Subject Classification}: Primary 11N37. \noindent \emph{Keywords: } gcd-sum function, regular integers modulo $n$, Riemann hypothesis, short interval result. \bigskip \hrule \bigskip \noindent (Concerned with sequences \seqnum{A018804} and \seqnum{A176345}.) \bigskip \hrule \bigskip \vspace*{+.1in} \noindent Received January 25 2011; revised version received May 24 2011. Published in {\it Journal of Integer Sequences}, June 10 2011. \bigskip \hrule \bigskip \noindent Return to \htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}. \vskip .1in \end{document}