Asymptotic Estimates for a Class of Summatory Functions Original Research Article

We establish explicit expressions for bothPandEin ∑nless-than-or-equals, slantx a(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 ∑nless-than-or-equals, slantx nβa(n) when the real numberβis not too small, and for ∫x1 E(t) dtand ∑nless-than-or-equals, slantx E(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.