A Generalization of Epstein Zeta Functions

§ 1. An Epstein zeta function (in its simplest form) is a function represented by the Dirichlet's series where a 1 … a k are real and n 1 , n 2 , … n k run through integral values. The properties of this function are well known and the simplest of them were proved by Epstein 2, 3. The aim of this note is to define a general class of Dirichlet's series, of which the above can be viewed as an instance, and to discuss the problem of analytic continuation of such series.