Analytic Number Theory on Higher Rank Groups

This project concerns research in the theory of numbers, in particular questions related to the distribution of prime numbers. One of the most fundamental open questions in modern number theory is the Riemann hypothesis, which connects the distribution of prime numbers with the distribution of complex zeros of the generating zeta function, the Riemann zeta function. The Riemann hypothesis asserts that the non-integer zeros lie on a line. The Riemann Zeta function is an example of an L-function. This is a project to study zeroes of general L-functions. In more detail, a Siegel zero is a type of potential counterexample to the (generalized) Riemann hypothesis on the zeros of Dirichlet L-functions connected to the study of the distribution of primes in arithmetic progressions. In the main part of this project, based on Sarnak's method, a novel technique will be used to derive the standard zero-free regions for general L-functions. This new technique combines the use of Maass-Selberg relations on higher rank groups which is expected to remove the technical sieve arguments in Sarnak's work. The new technique will also shed light on the classical Siegel zero problem as mentioned above. Other projects include the study of equidistribution of Maass forms on higher rank groups and spectral identities for L-functions. These will provide new bridges between analytic number theory, representation theory and ergodic theory.