New aspects of automorphic and smooth-automorphic forms

Number theory is likely to be one of the oldest disciplines of mathematics and hence of the human mind. Still, many of its biggest mysteries remain unsolved. And this despite the fact that number theory basically asks only one fundamental question, that, in a language, which is borrowed from natural sciences, could be formulated as follows: What are the laws of nature, that govern the interplay of all the natural numbers 1, 2, 3, 4, 5, 6, (and so forth to infinity)? One of the earliest discoveries of number theory, and hence, one of the oldest examples of such a law of nature, is that all these numbers can be written, in terms of multiplication, as the product of prime numbers revealing that prime numbers represent nothing different to mathematics than what atoms represent in physics and chemistry. However, whereas the number of chemical elements, i.e., different types of atoms, is finite, we know since the time of Euclid ( 300 B.C.) that the number of primes is in fact infinite: For every, even for the most horrendously huge looking prime number, there is always another prime number, which is even greater. And so forth to infinity! Needless to say, that the infinity of prime numbers causes a more than fundamental problem in every attempt that tries to understand them. The 20th century has made incredible progress in the more than 2500 years old history of trying to understand the distribution of primes in the infinite sea of all numbers: The most powerful of our todays techniques circle around what one calls automorphic forms. These automorphic forms are expected to encode all the information we want to have about primes, but also beyond, about other deep unsolved questions of number theory. In this research-project we are going to shade new light on some of the most fundamental problems arising in the study of automorphic forms: (i) We are going to investigate a new and more modern approach to the very fundaments of the theory, which aims to extend and improve the sheer notion of an automorphic form; (ii) we are going to set up an improved concept of Eisenstein series, which are amongst the most central objects in the realm of automorphic forms; (iii) we are going to establish new results on the values of the Riemann zeta-function and its interconnection with the dimension of certain spaces of automorphic forms.