In recent years, the moments of L-functions has been a topic of growing interest in the field of analytic number theory. New techniques, including applications of Random Matrix Theory and multiple Dirichlet series, have lead to many well-posed theorems and conjectures for the moments of various L-functions. In this thesis, we theoretically and numerically examine the integral moments of quadratic Dirichlet $L$-functions. In particular, we exhibit and discuss the conjectures for the moments which result from the applications of Random Matrix Theory, number theoretic heuristics, and the theory of multiple Dirichlet series. In the case of the cubic moment, we further numerically investigate the possible existence of additional lower order main terms.

In recent years, the moments of L-functions has been a topic of growing interest in the field of analytic number theory. New techniques, including applications of Random Matrix Theory and multiple Dirichlet series, have lead to many well-posed theorems and conjectures for the moments of various L-functions. In this thesis, we theoretically and numerically examine the integral moments of quadratic Dirichlet $L$-functions. In particular, we exhibit and discuss the conjectures for the moments which result from the applications of Random Matrix Theory, number theoretic heuristics, and the theory of multiple Dirichlet series. In the case of the cubic moment, we further numerically investigate the possible existence of additional lower order main terms.

<p>Arising from permutation representations of finite groups, Brauer-Kuroda relations are relations between Dedekind zeta functions of certain intermediate fields of a Galois extension of number fields. Let E be a totally real number field and let n ≥ 2 be an even integer. Taking s = 1 − n in the Brauer-Kuroda relations then gives a correspondence between orders of certain motivic and Galois cohomology groups. Following the works of Voevodsky and Wiles (cf. [33], [36]), we show that these relations give a direct relation on the motivic cohomology groups, allowing one to easily compute the higher class numbers, the orders of these motivic cohomology groups, of fields of high degree over Q from the corresponding values of its subfields. This simplifies the process by restricting the computations to those of fields of much smaller degree, which we are able to compute through Sage ([30]). We illustrate this with several extensive examples.</p> / Master of Science (MSc)

In this thesis we provide a body of knowledge that concerns Riemann zeta-function and its generalizations in a cohesive manner. In particular, we have studied and mentioned some recent results regarding Hurwitz and Lerch functions, as well as Dirichlet's L-function. We have also investigated some fundamental concepts related to these functions and their universality properties. In addition, we also discuss different formulations and approaches to the proof of the Prime Number Theorem and the Riemann Hypothesis. These two topics constitute the main theme of this thesis. For the Prime Number Theorem, we provide a thorough discussion that compares and contrasts Norbert Wiener's proof with that of Newman's short proof. We have also related them to Hadamard's and de la Vallee Poussin's original proofs written in 1896. As far as the Riemann Hypothesis is concerned, we discuss some recent results related to equivalent formulations of the Riemann Hypothesis as well as the Generalized Riemann Hypothesis.