Central values of derivatives of Dirichlet L-functions

TL;DR

This paper demonstrates that for large modulus q and derivative order k, the k-th derivatives of Dirichlet L-functions at the central point are non-zero for almost all even primitive characters, using mollifier techniques.

Contribution

It introduces a method to show non-vanishing of high derivatives of Dirichlet L-functions at the central point for almost all characters, extending previous results.

Findings

Non-vanishing of L^{(k)}(1/2,chi) for almost all chi in C(q,+)

Applicability of mollifier method to derivatives of L-functions

Results hold for large q and k

Abstract

Let C(q,+) be the set of even, primitive Dirichlet characters (mod q). Using the mollifier method we show that L^{(k)}(1/2,chi) is not equal to zero for almost all the characters chi in C(q,+) when k and q are large. Here, L^{(k)}(s,chi) is the k-th derivative of of the Dirichlet L-function L(s,chi).