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The Dedekind Zeta Function and the Class Number Formula Math 254 B Final Paper

The aim of this paper is to prove that the Dedekind zeta function for a number field has a meromorphic continuation to the complex plane, obtaining the analytic class number formula with it, and to present some of the applications of these results. In particular, It will be shown its use in proving Dirichlet’s prime number theorem and in calculating the class number of quadratic fields. The proof of the class number formula and analytic continuation of the zeta function will be complete except for some parts involving technical calculations which would not add much to the number theory concepts with which this paper tries to deal. It will mainly follow the proof in the book by Neukirch [2]. We will begin by demonstrating the use of the class number formula to find an expression for the class number of a quadratic field and prove Dirichlet’s prime number theorem. The proof of the main result will come after this.