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 ...

1. Differentiable Manifolds 2 2. Complex Manifolds 3 2.1. Riemann Surfaces of Genus One 4 2.2. Constructing Riemann Surfaces as Curves in P 6 2.3. Constructing Riemann Surfaces as Covers 9 2.4. Constructing Riemann Surfaces by Glueing 10 3. Topological Vector Bundles 11 3.1. The Tangent and Cotangent Bundles 13 3.2. Interlude: Categories, Complexes and Exact Sequences 14 3.3. Metrics on Vector ...

We are solving the classical Riemann-Hilbert problem of rank N > 1 on the extended complex plane punctured in 2m + 2 points, for N × N quasi-permutation monodromy matrices. Our approach is based on the finite gap integration method applied to study the Riemann-Hilbert by Kitaev and Korotkin [1], Deift, Its, Kapaev and Zhou [2] and Korotkin, [3]. This permits us to solve the Riemann-Hilbert prob...

The Hamiltonian of a quantum mechanical system has an affiliated spectrum. If this spectrum is the sequence of prime numbers, a connection between quantum mechanics and the nontrivial zeros of the Riemann zeta function can be made. In this case, the Riemann zeta function is analogous to chaotic quantum systems, as the harmonic oscillator is for integrable quantum systems. Such quantum Riemann z...

in this paper, we consider weighted composition operators betweenmeasurable differential forms and then some classic properties of these operators are characterized.

A Riemann surface is associated with its Jacobian through the Abel map. Using a plane algebraic curve representation of the Riemann surface, we provide an algorithm for the numerical computation of the Abel map. Since our plane algebraic curves are of arbitrary degree and may have arbitrary singularities, the Abel map of any connected compact Riemann surface may be obtained in this way.

Using a result of Harer, we prove certain upper bounds for the homotopical/cohomological dimension of the moduli spaces of Riemann surfaces of compact type, of Riemann surfaces with rational tails and of Riemann surfaces with at most k rational components. These bounds would follow from conjectures of Looijenga and Roth-

In this article we derive a generalization of the Riemann-Siegel asymptotic formula for the Riemann zeta function. By subtracting the singularities closest to the critical point we obtain a significant reduction of the error term at the expense of a few evaluations of the error function. We illustrate the efficiency of this method by comparing it to the classical Riemann-Siegel formula.

In this paper, we give both lower and upper bound estimates of the Bergman kernel for a degeneration of Riemann surfaces with constant curvature −1. As a result, we give a geometric proof of the Riemann–Rock theorem for a singular Riemann surface. Mathematics Subject Classification (2010). Primary 53C55; Secondary 32S30.