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This thesis is divided into three parts. In the first part we study the uniformity in the shifts in the asymptotic formulae for the second moment of the Riemann zeta-function and the first moments of the Hecke and the quadratic Dirichlet L-functions. In the second part we investigate the period function of the Eisenstein series. We use our results to give a simple proof of the Voronoi formula and to prove an exact formula for the second moment of the Riemann zeta function. Moreover, we study a family of cotangent sums, functions defined over the rationals, that generalize the Dedekind sum and share with it the property of satisfying a reciprocity formula. In the third part, we find optimal Dirichlet polynomials for the Nyman- Beurling criterion for the Riemann- hypothesis, conditionally on some separa- tion condition on the zeros of ((8) and on the Riemann hypothesis.

Bounded Schur multipliers were introduced and characterised several decades ago, and various applications of this algebra of functions have been discovered. More recently, research into different classes of unbounded multipliers has been carried out. In this thesis the theory of one such class, that of the local Schur multipliers, is extended in different settings. A dilation of minimal Stinespring representations of completely positive, bimodular maps on spaces of compact operators is obtained, and used to establish an unbounded version of Stinespring's Theorem. This theorem is applied to obtain a characterisation of positive local Schur multipliers. In addition, a relation is demonstrated between operator monotone functions and positive local Schur multipliers, and a description is given of positive multipliers of Toeplitz type. The theory of local multipliers is extended to the multidimensional setting, and a characterisation of such functions is obtained. Local operator multipliers are introduced as a non-commutative e analogue of local Schur multipliers and a description is provided, extending previously known results concerning completely bounded operator multipliers. Positive multipliers are defined in this setting) and characterised using elements of canonical positive cones. The two-dimensional Fourier algebra A2(G) of a compact, abelian group G is considered, and a number of results are obtained concerning the Arens product on its dual, VN(G) ®uh VN(G). It is shown that A2 (G) may be viewed as a left VN(G) ®ub VN(G)-module, and thus certain results of Eyroard are extended to the two-dimensional setting, leading to the establishment of a condition equivalent to the homeomorphic identification of the Gelfand spectrum of A2(G) with C2 .

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Chapter 1 reviews better-known factorization theorems of a square matrix. For example, a square matrix over a field can be expressed as a product of two symmetric matrices; thus square matrices over real numbers can be factorized into two diagonalizable matrices. Factorizing matrices over complex num hers into Hermitian matrices is discussed. The chapter concludes with theorems that enable one to prescribe the eigenvalues of the factors of a square matrix, with some degree of freedom. Chapter 2 proves that a square matrix over arbitrary fields (with one exception) can be expressed as a product of two diagona lizab le matrices. The next two chapters consider decomposition of singular matrices into Idempotent matrices, and of nonsingutar matrices into Involutions. Chapter 5 studies factorization of a comp 1 ex matrix into Positive-( semi )definite matrices, emphasizing the least number of such factors required / Mathematical Sciences / M.Sc. (MATHEMATICS)

Chapter 1 reviews better-known factorization theorems of a square matrix. For example, a square matrix over a field can be expressed as a product of two symmetric matrices; thus square matrices over real numbers can be factorized into two diagonalizable matrices. Factorizing matrices over complex numbers into Hermitian matrices is discussed. The chapter concludes with theorems that enable one to prescribe the eigenvalues of the factors of a square matrix, with some degree of freedom. Chapter 2 proves that a square matrix over arbitrary fields (with one exception) can be expressed as a product of two diagonalizable matrices. The next two chapters consider decomposition of singular matrices into Idempotent matrices, and of nonsingular matrices into Involutions. Chapter 5 studies factorization of a complex matrix into Positive-(semi)definite matrices, emphasizing the least number of such factors required. / Mathematical Sciences / M. Sc. (Mathematics)

Which matrices can be written as sums or products of square-zero matrices? This question is the central premise of this dissertation. Over the past 25 years a signi - cant body of research on products and linear combinations of square-zero matrices has developed, and it is the aim of this study to present this body of research in a consolidated, holistic format, that could serve as a theoretical introduction to the subject. The content of the research is presented in three parts: rst results within the broader context of sums and products of nilpotent matrices are discussed, then products of square-zero matrices, and nally sums of square-zero matrices. / Mathematical Sciences / M. Sc. (Mathematics)

Chapter 1 reviews better-known factorization theorems of a square matrix. For example, a square matrix over a field can be expressed as a product of two symmetric matrices; thus square matrices over real numbers can be factorized into two diagonalizable matrices. Factorizing matrices over complex num hers into Hermitian matrices is discussed. The chapter concludes with theorems that enable one to prescribe the eigenvalues of the factors of a square matrix, with some degree of freedom. Chapter 2 proves that a square matrix over arbitrary fields (with one exception) can be expressed as a product of two diagona lizab le matrices. The next two chapters consider decomposition of singular matrices into Idempotent matrices, and of nonsingutar matrices into Involutions. Chapter 5 studies factorization of a comp 1 ex matrix into Positive-( semi )definite matrices, emphasizing the least number of such factors required / Mathematical Sciences / M.Sc. (MATHEMATICS)