Research Work
Research Interest
JHU Number Theory Seminar
Applications of multiple Dirichlet series in mean values of L-functions, in “Multiple Dirichlet Series, Automorphic Forms, and Analytic Number Theory”, Proceedings of Symposia in Pure Mathematics 75, AMS, 2006, 43—58.
Special values of L-functions provide links between such diverse areas of mathematics as number theory, algebraic geometry, representation theory, topology and mathematical physics, and have been the subject of much interest and study. One successful approach is through the study of their average behaviors (the ``mean value problem''). In this expository paper, we overview some recent developments in the study of mean values of L-functions by applying the philosophy of multiple Dirichlet series.
Integral mean values of Maass L-functions, Int. Math. Res. Not., 2006, 41417, 1—19.
We consider the mean squares of L-functions associated to Maass forms with respect to Hecke congruence subgroups. In particular, we express the mean value as an inner product, and so avoid delicate discussions of generalized additive divisor problems.
On the cubic moment of quadratic Dirichlet L-functions, Math. Res. Letters, 12 (2005), no. 3, 413—424.
We consider the cubic moment of quadratic Dirichlet L-functions, and give the heuristic argument for the existence of some "exceptional main term" of order x^(3/4) in the asymptotic formula.
Integral mean values of modular L-functions, Journal of Number Theory, 115 (2005), no. 1, 100—122.
We consider the mean squares of L-functions associated to modular forms with respect to Hecke congruence subgroups, expressing the mean value as an inner product. This avoids the discussion of generalized additive divisor problems. As applications, we obtain asymptotic formulas for both weighted and unweighted mean squares.