Moments of L-functions and the Liouville-Green method

Olga Balkanova (University of Turku)

Frank Adams 1,

Various problems in number theory can be formulated in terms of central values of L-functions. In this talk, we consider the family of Hecke L-functions of fixed level and large weight. For this family, we show that the proportion of non-vanishing central L-values is at least 20% for an individual weight and at least 50% on average. The last result with an extra average over weight was first established by Iwaniec and Sarnak in 2000. Furthermore, they showed that any improvement over 50% would imply the non-existence of Landau-Siegel zeros for Dirichlet L-functions of real primitive characters. The main difficulty in proving the non-vanishing results for an individual weight is asymptotic evaluation of special functions appearing in exact formulas for moments of Hecke L-functions. We solve this problem using the Liouville-Green method, originating from the theory of approximation of second order differential equations. This is a joint work with Dmitry Frolenkov.
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