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Let m and n be positive integers, and the M"bius function. And let S f(m,n) be the function defined by

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\Sigma _{d|(m,n)} d\mu (m/d)f(n/d)$$ \end{document}
, where f is an arithmetical function. We show that this function has many properties like the Ramanujan sum. Firstly we study the partial summation formula involving S f(m,n) and taking f=, we obtain the Dirichlet series with the coefficients S(m,n) and S(m,n)d(m). Moreover we show a certain property which is analogous to the orthogonality relation of the Ramanujan sums.

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