Stark's Conjectures were formulated in the late 1970s and early 1980s. The most general version predicts that the leading coe cient of the Maclaurin series of an Artin L-function should be the product of an algebraic number and a regulator made up of character values and logarithms of absolute values of units. When known, Stark's conjecture provides a factorization of the analytic class number formula of Dirichlet. Stark succeeded in formulating a \re ned abelian" version of his conjecture when the L-function in question has a rst order zero and is associated with an abelian extension of number elds. In the spirit of Stark, Rubin and Popescu formulated analogous \re ned abelian" conjectures for Artin L-Functions which vanish to arbitrary order r at s = 0. These conjectures are identical to Stark's own re ned abelian conjecture when restricted to order of vanishing r = 1. We introduce Popescu's Conjecture C(L=F; S; r): We prove Popescu's Conjecture for multiquadratic extensions when the set of primes S of the base eld is minimal given minor restrictions on the S-class group of the base eld. This extends the results of Sands to the case where #S = r + 1. We present three in nite families of settings where our methods allow us to verify Popescu's conjecture. We formulate a conjecture that predicts when a fundamental unit of a real quadratic eld must become a square in a multiquadratic extension.

by Song Li-Min. / Thesis (M.Ph.)--Chinese University of Hong Kong, 1987. / Bibliography: leaves 175-178.

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Leung, Fung Bun. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 36). / Abstracts in English and Chinese. / Abstract --- p.i / Acknowledgement --- p.iii / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Tiling of Integers --- p.3 / Chapter 2.1 --- General Assumptions --- p.3 / Chapter 2.2 --- Periodicity of Tiling --- p.4 / Chapter 2.3 --- Cyclotomic Polynomials --- p.5 / Chapter 2.4 --- Equivalence --- p.10 / Chapter 3 --- Theorems of Integer Tiles --- p.12 / Chapter 3.1 --- Main Theorems --- p.12 / Chapter 3.2 --- Jusifications on Assumptions --- p.12 / Chapter 3.3 --- Proofs of Main Theorems --- p.14 / Chapter 3.3.1 --- Proof of Theorem 3.1.1 --- p.14 / Chapter 3.3.2 --- Proof of Theorem 3.1.2 --- p.15 / Chapter 3.3.3 --- Proof of Theorem 3.1.3 --- p.15 / Chapter 4 --- Structure of Integer Tiles --- p.20 / Chapter 4.1 --- Classification --- p.20 / Chapter 4.2 --- Structure of Tiles with Cardinality pα --- p.22 / Chapter 4.3 --- Structure of Tiles with Cardinality pαqβ --- p.23 / Chapter 4.4 --- More Examples --- p.25 / Chapter 5 --- Complementing Pairs Mod pqr --- p.26 / Chapter 5.1 --- Introduction --- p.26 / Chapter 5.2 --- A Conjecture and Proofs --- p.27 / Chapter 5.3 --- A Summary of Our Results --- p.29 / Chapter A --- Number Systems --- p.30 / Chapter A.l --- Definitions --- p.30 / Chapter A.2 --- Structure of tiles of Z+ and Nn --- p.31 / Chapter B --- Notations --- p.35

Let ω(m) be the number of distinct prime factors of m. A celebrated theorem of Erdös-Kac states that the quantity (ω(m)-loglog m)/√(loglog m) distributes normally. Let φ(m) be Euler's φ-function. Erdös and Pomerance proved that the quantity(ω(φ(m)-(1/2)(loglog m)^2)\((1/√(3)(loglog m)^(3/2)) also distributes normally. In this thesis, we prove these two results. We also prove a function field analogue of the Erdös-Pomerance Theorem in the setting of the Carlitz module.

Let ω(m) be the number of distinct prime factors of m. A celebrated theorem of Erdös-Kac states that the quantity (ω(m)-loglog m)/√(loglog m) distributes normally. Let φ(m) be Euler's φ-function. Erdös and Pomerance proved that the quantity(ω(φ(m)-(1/2)(loglog m)^2)\((1/√(3)(loglog m)^(3/2)) also distributes normally. In this thesis, we prove these two results. We also prove a function field analogue of the Erdös-Pomerance Theorem in the setting of the Carlitz module.

Roth's theorem, proved by Roth in 1953, states that when A is a subset of the integers [1,N] with A dense enough, A has a three term arithmetic progression (3-AP). Since then the bound originally given by Roth has been improved upon by number theorists several times. The theorem can also be generalized to finite abelian groups. In 1994 Meshulam worked on finding an upper bound for subsets containing only trivial 3-APs based on the number of components in a finite abelian group. Meshulam’s bound holds for finite abelian groups of odd order. In 2003 Lev generalised Meshulam’s result for almost all finite abelian groups. In 2009 Liu and Spencer generalised the concept of a 3-AP to a linear equation and obtained a similar bound depending on the number of components of the group. In 2011, Liu, Spencer and Zhao generalised the 3-AP to a system of linear equations. This thesis is an overview of these results.

Thesis (doctoral)--Georg-August-Universität zu Göttingen, 1913. / Cover title. Vita.

Thesis (doctoral)--Georg-Augusts-Universität zu Göttingen, 1910. / Vita. Includes bibliographical references.