On a Weighted Mean Square Result of the Error Terms of Some Arithmetical Functions Original Research Article

Let a(n) be an arithmetical function satisfying some conditions and write ∑′nless-than-or-equals, slantx a(n)=main term+E(x). We obtain an asymptotic formula for a weighted mean square of the error termimage for some constants σ0 and c, by modifying a method that seems due to Titchmarsh. This method utilizes the following tools: the Perron formula, the Parseval Theorem, and a Tauberian Theorem. The main difference between the present and the original method is that we include the contribution of the poles of the associated Dirichlet series on a certain line in the main term.