Families of L-Functions and Analytic Number Theory

This research project centers on L-functions, which are mathematical objects useful for studying a broad array of questions in number theory. An L-function is a special kind of function that packages together information about an arithmetical object that arises from studying it modulo p for each prime p. For instance, the Riemann zeta function is the simplest example of an L-function and has proven to be indispensable in studying the distribution of the prime numbers. Other types of L-functions are crucial for understanding if certain polynomial equations have solutions, or more generally, how many solutions there are. Often these questions are related to how large the L-function is at a special point. Much of this project concerns the development of new tools for studying how L-functions may fit into families and using these tools to better understand individual L-functions. The investigator will continue to advise PhD students and to mentor and collaborate with undergraduate students, especially through the Texas A&M Research Experience for Undergraduates. This type of mentorship is invaluable in preparing students for graduate studies, particularly for undergraduate students from non-PhD granting institutions as well as from population groups underrepresented in STEM fields. The project will study new families of L-functions and use them as tools for estimating L-functions on the critical line. In particular, the work aims to develop new large sieve inequalities, which are flexible tools broadly useful in analytic problems on L-functions. The investigator and his students will study moments of L-functions in smaller sub-families than have previously been considered. Another line of work concerns new variants on the quantum unique ergodicity problem, which connects families of L-functions to properties of automorphic forms. With his undergraduate students, the investigator will study properties of generalized Dedekind sums. The methods employed will be techniques from analytic number theory such as summation formulas, functional equations, exponential sums and integrals, and the spectral theory of automorphic forms, including the Arthur-Selberg trace formula and the relative trace formula.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.

This research project centers on L-functions, which are mathematical objects useful for studying a broad array of questions in number theory. An L-function is a special kind of function that packages together information about an arithmetical object that arises from studying it modulo p for each prime p. For instance, the Riemann zeta function is the simplest example of an L-function and has proven to be indispensable in studying the distribution of the prime numbers. Other types of L-functions are crucial for understanding if certain polynomial equations have solutions, or more generally, how many solutions there are. Often these questions are related to how large the L-function is at a special point. Much of this project concerns the development of new tools for studying how L-functions may fit into families and using these tools to better understand individual L-functions. The investigator will continue to advise PhD students and to mentor and collaborate with undergraduate students, especially through the Texas A&M Research Experience for Undergraduates. This type of mentorship is invaluable in preparing students for graduate studies, particularly for undergraduate students from non-PhD granting institutions as well as from population groups underrepresented in STEM fields. The project will study new families of L-functions and use them as tools for estimating L-functions on the critical line. In particular, the work aims to develop new large sieve inequalities, which are flexible tools broadly useful in analytic problems on L-functions. The investigator and his students will study moments of L-functions in smaller sub-families than have previously been considered. Another line of work concerns new variants on the quantum unique ergodicity problem, which connects families of L-functions to properties of automorphic forms. With his undergraduate students, the investigator will study properties of generalized Dedekind sums. The methods employed will be techniques from analytic number theory such as summation formulas, functional equations, exponential sums and integrals, and the spectral theory of automorphic forms, including the Arthur-Selberg trace formula and the relative trace formula.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.