Non-vanishing of central values of L-functions with angular restrictions

We study the angular restrictions for the second moment of toroidal families of $L$-functions using the general theory of trace functions. With the mollification technique we deduce non-vanishing of a positive proportion. Our two main ingredients are classification results of Katz to determine the sheaves at play and a recent result of Fouvry, Kowalski, Michel and Sawin to bound bilinear sums of their trace functions.

💡 Research Summary

The paper investigates the simultaneous non‑vanishing of central values of Dirichlet L‑functions attached to primitive characters, under the additional constraint that the argument (or “angle”) of the normalized Gauss sum ε(χ)=e^{iθ(χ)} lies in a prescribed interval I⊂(−π,π]. Building on earlier work of Fouvry, Kowalski and Michel (FKM24), which computed the second moment
\