Analytical Methods

Abstract

In this section we introduce the Dirichlet series defining the Dedekind zeta-function, and also some other kinds of zeta-functions, including Dirichlet’s L-functions, and derive the functional equations for them. Our arguments will be based on the results of Chap. 6. Subsequent sections are devoted to asymptotic distribution of ideals and prime ideals. We shall use the tauberian theorem of Delange, an account of which is given in Appendix II, as well as complex integration in its simplest form. We adopt the convention that Σ _I and Σ_p denote summations over all non-zero ideals, respectively all non-zero prime ideals of the considered algebraic number field. We shall also denote² by σ,t the real, respectively the imaginary part of the complex variable s.