On a weighted mean square result of the error terms of some arithmetical functions

Abstract

Let a(n) be an arithmetical function satisfying some conditions and write ∑′n≤xa(n)=main term+E(x). We obtain an asymptotic formula for a weighted mean square of the error term∫1T|E(y)|2y2σ 0 +1dy∼clogT 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. © 2000 Academic Press.