Abstract

This thesis aims to extend and elaborate on the initial sections of Neal Koblitz's article titled "A New Proof of Certain Formulas for p-Adic L-Functions". Koblitz's article focuses on the construction of p-adic L-functions associated with Dirichlet's character and the computation of their values at s = 1. He employs measure-theoretic methods
to construct the p-adic L-functions and compute the Leopoldt formula L_p(1,χ).

To begin, we devote the first section (1.1) to providing comprehensive proof of Dirichlet's theorem for prime numbers. This is done because the theorem serves as a noteworthy example of how Dirichlet L-functions became relevant in the field of Number Theory.

In the second chapter, we introduce the complex version of Dirichlet L-functions and Riemann Zeta functions. We explore their analytical properties, such as functional equations and analytic continuation. Subsequently, we construct the field of p-adic numbers and equip it with the p-adic norm to facilitate analysis. We introduce measures
and perform p-adic integrations.

Finally, we delve into the concept of p-adic interpolation for the Riemann Zeta function, aiming to establish the p-adic Zeta function. To accomplish this, we employ Mazur's measure-theoretic approach, utilizing the tools introduced in the third chapter. The thesis concludes by incorporating Koblitz's work on this subject.