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\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} \\
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\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}


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