Probabilistic methods in analysis and number theory

In this project we will investigate some specific mathematical problems from number theory, analysis and numerical mathematics, whose common feature is that probabilistic methods form an important tool for their investigation. One part of the project will focus on theoretical problems in uniform distribution theory, analytic number theory, and on the convergence of function series. The second part will focus on problems from applied mathematics, and in particular from tractability theory, such as problems concerning the error of the quasi-Monte Carlo method for the numerical integration of high-dimensional functions. Some of the main aims of the project are the following: - To use a strong new technique, developed recently by the proposer together with I. Berkes and K. Seip, to attack long-standing open problems from metric number theory and problems concerning the convergence of series of dilated functions. - To investigate the connection between the sums involving greatest common divisors, which are the central element of the technique mentioned above, with the Riemann zeta function, and to use these GCD sums and the resonance method to prove new lower bounds for the maximum of the Riemann zeta function - To attack the problems of the inverse of the discrepancy and of the construction of low-discrepancy point sets of moderate cardinality in high dimensions. These problems are crucial, as they decide about the feasibility or infeasibility of high-dimensional Quasi-Monte Carlo integration. - To prove results about the existence of low-discrepancy point sets with respect to non-uniform measures, and to find construction or transformation methods for such point sets. Such results will allow to apply the Quasi-Monte Carlo method to important new classes of functions, which are out of reach for the standard method using the uniform measure on the unit cube. Overall, in this project we will obtain results which provide valuable new insight into important open problems, both in theoretical and applied mathematics, and we will develop new probabilistic methods for the investigation of problems in analysis, number theory and numerical analysis.

This research project was concerned with applications of probability theory in other mathematical areas such as analysis, number theory, and numerical analysis. Among the main research topics were the theory of the Riemann zeta function, local distribution properties of arithmetic sequences, and Diophantine Approximation. During the course of the research project, several unexpected connections were observed, for example to the Berry-Tabor conjecture in theoretical physics or the theory of quantum modular forms of Zagier. At any time of the project, two PostDoc researchers were working in the project (additionally to the project leader). Two PhD students were educated during the course of the project.