510 Mathematik
Refine
Has Fulltext
- yes (3)
Is part of the Bibliography
- yes (3)
Year of publication
- 2013 (3) (remove)
Document Type
- Doctoral Thesis (3) (remove)
Language
- English (3) (remove)
Keywords
Institute
The Riemann zeta-function forms a central object in multiplicative number theory; its value-distribution encodes deep arithmetic properties of the prime numbers. Here, a crucial role is assigned to the analytic behavior of the zeta-function on the so called critical line. In this thesis we study the value-distribution of the Riemann zeta-function near and on the critical line. Amongst others we focus on the following.
PART I: A modified concept of universality, a-points near the critical line and a denseness conjecture attributed to Ramachandra.
The critical line is a natural boundary of the Voronin-type universality property of the Riemann zeta-function. We modify Voronin's concept by adding a scaling factor to the vertical shifts that appear in Voronin's universality theorem and investigate whether this modified concept is appropriate to keep up a certain universality property of the Riemann zeta-function near and on the critical line. It turns out that it is mainly the functional equation of the Riemann zeta-function that restricts the set of functions which can be approximated by this modified concept around the critical line.
Levinson showed that almost all a-points of the Riemann zeta-function lie in a certain funnel-shaped region around the critical line. We complement Levinson's result: Relying on arguments of the theory of normal families and the notion of filling discs, we detect a-points in this region which are very close to the critical line.
According to a folklore conjecture (often attributed to Ramachandra) one expects that the values of the Riemann zeta-function on the critical line lie dense in the complex numbers. We show that there are certain curves which approach the critical line asymptotically and have the property that the values of the zeta-function on these curves are dense in the complex numbers.
Many of our results in part I are independent of the Euler product representation of the Riemann zeta-function and apply for meromorphic functions that satisfy a Riemann-type functional equation in general.
PART II: Discrete and continuous moments.
The Lindelöf hypothesis deals with the growth behavior of the Riemann zeta-function on the critical line. Due to classical works by Hardy and Littlewood, the Lindelöf hypothesis can be reformulated in terms of power moments to the right of the critical line. Tanaka showed recently that the expected asymptotic formulas for these power moments are true in a certain measure-theoretical sense; roughly speaking he omits a set of Banach density zero from the path of integration of these moments. We provide a discrete and integrated version of Tanaka's result and extend it to a large class of Dirichlet series connected to the Riemann zeta-function.
The work at hand studies problems from Loewner theory and is divided into two parts:
In part 1 (chapter 2) we present the basic notions of Loewner theory. Here we use a modern form which was developed by F. Bracci, M. Contreras, S. Díaz-Madrigal et al. and which can be applied to certain higher dimensional complex manifolds.
We look at two domains in more detail: the Euclidean unit ball and the polydisc. Here we consider two classes of biholomorphic mappings which were introduced by T. Poreda and G. Kohr as generalizations of the class S.
We prove a conjecture of G. Kohr about support points of these classes. The proof relies on the observation that the classes describe so called Runge domains, which follows from a result by L. Arosio, F. Bracci and E. F. Wold.
Furthermore, we prove a conjecture of G. Kohr about support points of a class of biholomorphic mappings that comes from applying the Roper-Suffridge extension operator to the class S.
In part 2 (chapter 3) we consider one special Loewner equation: the chordal multiple-slit equation in the upper half-plane.
After describing basic properties of this equation we look at the problem, whether one can choose the coefficient functions in this equation to be constant. D. Prokhorov proved this statement under the assumption that the slits are piecewise analytic. We use a completely different idea to solve the problem in its general form.
As the Loewner equation with constant coefficients holds everywhere (and not just almost everywhere), this result generalizes Loewner’s original idea to the multiple-slit case.
Moreover, we consider the following problems:
• The “simple-curve problem” asks which driving functions describe the growth of simple curves (in contrast to curves that touch itself). We discuss necessary and sufficient conditions, generalize a theorem of J. Lind, D. Marshall and S. Rohde to the multiple-slit equation and we give an example of a set of driving functions which generate simple curves because of a certain self-similarity property.
• We discuss properties of driving functions that generate slits which enclose a given angle with the real axis.
• A theorem by O. Roth gives an explicit description of the reachable set of one point in the radial Loewner equation. We prove the analog for the chordal equation.