The Dirichlet problem

Thesis (M.A.)--Boston UniversityThe problem of finding the solution to a general eliptic type partial differential equation, when the boundary values are given, is generally referred to as the Dirichlet Problem. In this paper I consider the special eliptic equation of ∇2 J=0 which is Laplace's equation, and I limit myself to the case of two dimensions. Subject to these limitations I discuss five proofs for the existence of a solution to Laplace's equation for arbitrary regions where the boundary values are given. [TRUNCATED

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

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

Lectures on Functional Theory and Partial Differential Equations

Paper represents a summary of three lectures delivered in Anderson Hall on April 25, 26, 29, 196

The Riemann zeta function

The Riemann Zeta Function has a deep connection with the distribution of primes. This expository thesis will explain the techniques used in proving the properties of the Rieman Zeta Function, its analytic continuation to the complex plane, and the functional equation that the the Riemann Zeta Function satisfies

On the order of Dirichlet L-functions

Let L(s, X) be a Dirichlet L-function, where X is a nonprincipal character (mod q) and s = σ + it

Distributions: The evolution of a mathematical theory

AbstractThe theory of distributions, or generalized functions, evolved from various concepts of generalized solutions of partial differential equations and generalized differentiation. Some of the principal steps in this evolution are described in this paper

Dirichlet series as a generalization of power series

M.S.James M. Osbor

An explicit formula for Dirichlet\u27s L-Function

The Riemann zeta function has a deep connection to the distribution of primes. In 1911 Landau proved that, an explicit formula where ρ = β + iγ denotes a complex zero of the zeta function and Λ(x) is an extension of the usual von Mangoldt function, so that Λ(x) = log p if x is a positive integral power of a prime p and Λ(x) = 0 for all other real values of x. Landau’s remarkable explicit formula lacks uniformity in x and therefore has limited applications to the theory of the zeta function. In 1993 Gonek proved a version of Landau’s explicit formula that is uniform in both variables x and T. This explicit formula was used to estimate various sums involving the zeros of the zeta function, such as the discrete mean value formula for the zeta function. The purpose of this thesis is to obtain a generalization of Landau’s and Gonek’s explicit formulas in terms of the zeros of the Dirichlet L-function. To accomplish this, we employ the argument principal, Cauchy’s residue theorem, and an inequality of Selberg

Singular behavior of the Dirichlet problem in Hölder spaces of the solutions to the Dirichlet problem in a cone

In the present study we consider the solution of the Dirichlet problem in conical domain. For general elliptic problems in non Hilbertian Sobolev spaces built on $L^{p},1<p<\infty$ the theory of sums of operators developed by Dore-Venni $\left[8\right]$ provides an optimal result. Holder spaces, as opposed to LP spaces, are not UMD. Using the results of Da Prato-Grisvard $\left[6\right]$ and Labbas $\left[14\right]$ we cope with the singular behaviour of the solution in the framework of H$\ddot{\textrm{o}}$lder and little H$\ddot{\textrm{o}}$lder spaces

On the size of the fundamental unit of real quadratic fields

There exists a positive density of fundamental discriminants D such that the attached fundamental unit of the quadratic number field obtained by adding the square root of D is essentially greater than the cube of D