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Register a new userOn the Tong-type identity and the mean square of the error term for an extended Selberg class
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In 1956, Tong established an asymptotic formula for the mean square of the error term in the summatory function of the Piltz divisor function $d_3(n).$ The aim of this paper is to generalize Tongs method to a class of Dirichlet series that satisfy a functional equation. As an application, we can establish the asymptotic formulas for the mean square of the error terms for a class of functions in the well-known Selberg class. The Tong-type identity and formula established in this paper can be viewed as an analogue of the well-known Voronois formula.
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Let $ba=(a_1,a_2,ldots,a_k)$, where $a_j (j=1,ldots,k)$ are positive integers such that $a_1 leq a_2 leq cdots leq a_k$. Let $d(ba;n)=sum_{n_1^{a_1}cdots n_k^{a_k}=n}1$ and $Delta(ba;x)$ be the error term of the summatory function of $d(ba;n)$. In this paper we show an asymptotic formula of the mean square of $Delta(ba;x)$ under a certain condition. Furthermore, in the cases $k=2$ and 3, we give unconditional asymptotic formulas for these mean squares.
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