The origin of analytic properties of zeta functions on the domain where they are divergent

点击翻译按钮获取中文摘要

Let F(s) be a Dirichlet series associated with a sequence a(n). We studied the origin of analytic properties of F(s) from the point of view of an error term coming from sum of a(n).We prove that if m-tuple integrals of the error term are bounded by functions having mild orders, then F(s) can be continued analytically over the whole plane and satisfy a certain additional assumption. The converse assertion is also proved. We also study a relation between values of the m-tuple integrals of the error term and special values of a certain Dirichlet series.