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Abstract: Let m and n be positive integers, and the M"bius function. And let S f(m,n) be the function defined by $$\Sigma _{d|(m,n)} d\mu (m/d)f(n/d)$$ , 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.

Abstract. For $-1\leq a \leq 0$ let ${\it\Delta} _{a}(x;r)$ denote the error term in the asymptotic formula for $\sum \limits _{n \leq x} {\it\sigma} _{a}(n)e(rn)$. We study the mean square of the function $|{\it\Delta} _{a}(x+U;r)-{\it\Delta} _{a}(x;r)|$ for short intervals.