Value Distribution of Zeta- and L-Functions

The study of the Riemann zeta-function is undoubtedly among the main objects of research in the field of analytic number theory. This function has been first investigated by the Swiss mathematician Leonhard Euler who gave an alternative proof of Euclids well-known result that there are infinitely many prime numbers. Prime numbers are natural numbers who can be divided only by 1 and themselves and this distinctive property makes them very useful in real life applications, such as in cryptography. Therefore, understanding how they are distributed among the natural numbers has always been in the heart of mathematical research. It was the German mathematician Bernhard Riemann who improved significantly on the work of Euler and showed that there is a bidirectional relation between the value distribution of the Riemann zeta-function and the distribution of the prime numbers. In light of this fact, researchers studied extensively this function and its attributes over the years. Our aim is to focus on two aspects of the Riemann zeta-functions value-distribution, as well as to which extent comparable results hold for other functions related to the Riemann zeta -function. At first we study the universality property of the Riemann zeta-function which can be visualised roughly in the following way: if the (three-dimensional) graph of the norm of a complex-variable function is like the dunes of a sandy coast, then any dune of arbitrary size can be found sooner or later in the graph of the Riemann zeta-function with a given error. This simply means that the Riemann zeta-function approximates almost any other function at some point of time (hence the universality characterization). We will examine how soon and how frequently this phenomenon occurs. In particular, we will show that approximating a function with the Riemann zeta-function within a deterministic frame of moments in time is almost always possible. Moreover, it is known that in any given continuous time interval there are dunes in the graph of the Riemann zeta-function with increasing height. This is translated to the subject of extreme values of the Riemann zeta-function and in the second part of the project we will concentrate on determining how these heights behave in a given discrete time frame. Both topics will be examined also for a wider class of zeta- and L-functions which, as in the case of the Riemann zeta-function, entail information on specific arithmetical functions.

The main purpose of this project was to study various aspects of the value distribution of the Riemann zeta function. Understanding the behaviour of this complex-valued function can give us an insight into the distribution of prime numbers. In a sense, there is a certain duality between their distributions. We can say that the primes are distributed almost randomly among the integers. The same can be said of the graph of the zeta function along the vertical lines of the complex plane. For example, we have shown that any smooth plane curve can be approximated by the graph of the zeta function along only one of these lines. This motivated the study of the curvature of these graphs and how often they are close to a fixed point in the plane. Such questions are related to the universality theorem, which roughly states that the zeta function can approximate almost any target function and that this phenomenon repeats itself periodically. A natural question is to determine the frequency of these occurrences and how quickly they can happen, to which we have given partial answers. We have also obtained results on the extremal behavior of the zeta function along discrete sets. Proving upper bounds for the growth of the zeta function along vertical lines is a very challenging problem (ultimately related to the prime distribution, of course), and proving lower bounds is an indication of how far we are from reaching the conjectural bounds. Many of the above results were obtained by using tools from analytic number theory and harmonic analysis. The techniques developed have also allowed us to tackle problems related to the distribution of so-called Dedekind sums and the gap distribution of dilated monomial sequences. Dedekind sums were introduced by Dedekind in the context of modular forms and give information about the distribution of partial quotients of a rational number or, equivalently, about the number of steps of the Euclidean algorithm for division between two integers. We have shown that there is a certain bias in the distribution of these sums, which translates into saying that a variant of the Euclidean algorithm takes longer to complete for certain pairs of integers. The study of monomial sequences is related, among other things, to the so-called Berry-Tabor conjecture, which hypothesizes a link between the pseudo-randomness properties of energy levels and the dynamics of quantum systems. One way of understanding their gap distribution is to examine how the elements of such a sequence correlate with each other. There are very few examples for which we have a concrete understanding of these correlations, but during this project we were able to make a notable step in this direction.