ASYMPTOTIC ESTIMATES FOR A CLASS OF SUMMATORY FUNCTIONS

We establish explicit expressions for both P and E in Sigma(n less tha n or equal to x)a(n)= P(x) + E(x)= ''principal term''+''error term'', when the (complex) arithmetical function a has a generating function o f the form zeta(s) Z(s), where zeta is the Riemann zeta function, and where Z has a representation as a Dirichlet series having an abscissa of absolute convergence smaller than 1 land satisfying some other cond itions). We obtain O-estimates land in some cases Omega-estimates) on E. We also obtain asymptotic expressions for Sigma(n less than or equa l to x)n(beta)a(n) when the real number beta is not too small, and for integral(1)(x)E(t)dt and Sigma(n less than or equal to x)E(n). This c an 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. (C) 1998 Academi c Press.