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p-Adic L-Functions Attached to Dirichlet's Character


p-Adic L-Functions Attached to Dirichlet's Character

Shahabi, Mohammadhossein (2023) p-Adic L-Functions Attached to Dirichlet's Character. Masters thesis, Concordia University.

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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.

Divisions:Concordia University > Faculty of Arts and Science > Mathematics and Statistics
Item Type:Thesis (Masters)
Authors:Shahabi, Mohammadhossein
Institution:Concordia University
Degree Name:M. Sc.
Date:12 September 2023
Thesis Supervisor(s):Iovita, Adrian
ID Code:992936
Deposited By: Mohammadhossein Shahabi
Deposited On:16 Nov 2023 20:51
Last Modified:16 Nov 2023 20:51
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