Dirichlet Series and Polynomial Euler Products

In this chapter, we introduce a class of Dirichlet series satisfying several quite natural analytic axioms in addition with two arithmetic conditions, namely, a polynomial Euler product representation and some kind of prime number theorem. The elements of this class will be the main actors in the sequel; however, for some of the later results we do not need to assume all of these axioms. Further, we shall prove mean-value estimates for the Dirichlet series coe.cients of these L-functions as well as asymptotic mean-square formulae on vertical lines in the critical strip. These estimates will turn out to be rather useful in later chapters.