NSF-Analytic Number Theory and Periods of Automorphic Forms
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abstract

The principal investigator proposes several projects in analytic number theory concerning Fourier coefficients of modular forms and moments of L-functions. The investigator will combine spectral methods with formulas which relate these quantities to periods of automorphic forms to establish asymptotic formulas with strong bounds on the error terms. In project 1 the investigator will establish a new asymptotic formula for the Hardy-Ramanujan partition function and thus provide an alternative approach to computing large partition numbers. In project 2 the investigator will use traces of weak Maass forms to construct new examples of mock modular forms and establish asymptotic formulas for their coefficients. In project 3 the investigator will establish asymptotic formulas for moments of L-functions and apply these results to various non-vanishing and subconvexity problems. Modular forms and L-functions are objects of fundamental importance in number theory. For example, the Fourier coefficients of modular forms contain arithmetic and combinatorial information, while central values of L-functions are related to invariants of elliptic curves. The proposed research will result in a greater understanding of these objects and pave the way for new applications. One such application involves computing large partition numbers, which have important uses in scientific computing and cryptography. The proposed research will also lead to problems suitable for undergraduate students, graduate students, and post-doctoral fellows. Some of these problems will be studied by undergraduates advised by the investigator through the NSF Research Experiences for Undergraduates program.