Number Theory Seminar
Wednesday, December 7, 2022, 3:15pm, Physics 119
Caroline Turnage-Butterbaugh
Moments of Dirichlet L-functions
Abstract:
In recent decades there has been much interest and measured progress in the study of moments of L-functions. Despite a great deal of effort spanning over a century, asymptotic formulas for moments of L-functions remain stubbornly out of reach in all but a few cases. I will begin this talk by reviewing what is known for moments of the Riemann zeta-function on the critical line, and we will then consider the problem for the family of all Dirichlet L-functions of even primitive characters of bounded conductor. A heuristic of Conrey, Farmer, Keating, Rubenstein, and Snaith gives a precise prediction for the asymptotic formula for the general 2kth moment of this family. I will outline how to harness the asymptotic large sieve to prove an asymptotic formula for the general 2kth moment of approximations of this family. The result, which assumes the generalized Lindelöf hypothesis for large values of k, agrees with the prediction of CFKRS. Moreover, it provides the first rigorous evidence beyond the so-called “diagonal terms” in their conjectured asymptotic formula for this family of L-functions. This is joint work with Siegfred Baluyot and a product of the NSF Focused Research Group “Averages of L-functions and Arithmetic Stratification.”

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