Preâmbulos aritméticos: da função Zeta às fórmulas explícitas

Abstract

The main aim of this study is to begin the organization and presentation of the fundamental results of analytic number theory to the Portuguese language. I elaborate over the history of the theory in the context of the history of mathematics. Several subjects important to the understanding of the results are gathered. The Gamma function and its arithmetical importance are brie y analyzed. The Zeta function and the fundamental connection it establishes between natural and prime numbers is presented, as well as its evaluation on the integers and properties of these numbers. The location and relevance of its roots are brie y discussed. Dirichlet series are de ned, its basic properties and particular constructions are studied using the Zeta function. How analytic continuation and tauberian theorems provide results such as the Prime Number Theorem is shown. I will present the theories of Weierstrass and Hadamard on the representation of functions with explicit factors of their roots. I conclude the work by presenting the explicit formulae of number theory, which serve as nal guidance for most of the theory here developed.