L-functions and prime numbers

Kansas State University, Manhattan KS

Investigators

Abstract

Research in number theory is motivated by deep questions regarding the structure of the natural numbers. Such questions have fascinated mathematicians for millennia and have found applications in a number of diverse areas, e.g., cryptography, signal processing and data analysis, communications. Some of the most fundamental objects of study here are the prime numbers. The study of prime numbers uses tools from sieve theory, as well as connections to the zeros of L-functions. This award will enable the PI to further explore fundamental questions about families of L-functions and prime numbers. Further, this will support the PI as he continues to mentor graduate and undergraduate students, organizes conferences and seminars, and participates in extra-curricular outreach. The PI will study the distribution of values and zeros of L-functions. We study statistics over quadratic characters with fundamental discriminants, where prominent problems have intimate connections to arithmetic problems. We extend results of Conrey, Iwaniec and Soundararajan on the critical line theorem for Dirichlet L-functions by building on our previous joint work on moments. We study the eighth moment of automorphic L-functions, which has already exhibited novel features. We investigate the statistics of a large orthogonal family for the first time, where we have a plan to “reduce the conductor”; this also leads to simultaneous non-vanishing results. We attack subconvexity results for certain L-functions in conductor dropping families, which builds on our previous work on moments. Further, we study problems in counting prime numbers of a special form, which is connected to foundational conjectures in prime number theory. This project is jointly funded by Algebra and Number Theory Program 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.