Moments of Large Families of L-Functions and Related Questions

One of the most famous conjectures in mathematics is about zeros of the Riemann zeta function. This interest extends to the study of L-functions, which have connections with many diverse areas of mathematics such as harmonic analysis, random matrix theory and probability. In this area, there is a foundational heuristic that the distribution of values and zeros of L-functions should match analogous statistics from classical compact groups of random matrices.

It has led to a deep set of conjectures, which remain unresolved at many levels. Indeed, rigorous proofs of even special cases of these conjectures are almost non-existent. The proposed research will lead to a better understanding of these conjectures by proving special cases for certain large families of L-functions that have previously not been understood in this context.

The award will provide opportunities for research training and collaboration for graduate students and postdocs. The PI will also use the grant to organize number theory seminars and mentor students from underrepresented groups.

The aim of this award is to study statistics involving values and zeros of families of L-functions. To be precise, the PI will aim to prove new instances of the moment conjectures for high moments of large families of L-functions, as well as extend the current knowledge on the distribution of their zeros (e.g. n-level density). Moreover, attention will be paid to certain attractive applications of understanding of high moments, especially subconvexity bounds and simultaneous non-vanishing results.

The techniques employed include traditional Fourier analysis, spectral theory, and combinatorial number theory.

This project is jointly funded by the Algebra and Number Theory program in the Division of Mathematical Sciences and the Established Program to Stimulate Competitive Research (EPSCoR).

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.