On a relation between sums of arithmetical functions and dirichlet series

We introduce a concept called good oscillation. A function is called good oscillation, if its m-tuple integrals arbeounded by functions having mild orders. We prove that if the error terms coming from summatory functions of arithmetical functions are good oscillation, then the Dirichlet series associated with those arithmetical functions can be continued analytically over the whole plane. We also study a sort of converse assertion that if the Dirichlet series are continued analytically over the whole plane and satisfy a certain additional assumption, then the error terms coming from thesummatory functions of Dirichlet coefficients are good oscillation.