Asymptotic Estimates for a Class of Summatory Functions

AbstractWe establish explicit expressions for bothPandEin ∑n⩽xa(n)=P(x)+E(x)=“principal term”+“error term”, when the (complex) arithmetical function a has a generating function of the formζ(s)Z(s), whereζis the Riemann zeta function, and whereZhas a representation as a Dirichlet series having an abscissa of absolute convergence smaller than 1 (and satisfying some other conditions). We obtainO-estimates (and in some casesΩ-estimates) onE. We also obtain asymptotic expressions for ∑n⩽xnβa(n) when the real numberβis not too small, and for ∫x1E(t)dtand ∑n⩽xE(n). This can be applied to a number of arithmetical functions a which have been studied in the literature with various methods. In most cases what we obtain improves on, or extends, the existing results