FWF - VDoZaLF - 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 Euclid’s 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-function’s 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.