On optimality of mollifiers

Abstract

Mollifiers are used in a variety of contexts, for instance to study the non-vanishing of $L$-functions. In this paper, we study the general question of finding optimal mollifiers and provide criteria to identify them provided the corresponding mollified moments can be computed. As an application, we study the non-vanishing of central values of Dirichlet $L$-functions. In particular we show that the Michel-Vanderkam mollifier is optimal in a wide class of balanced two-piece mollifiers as well as provide a new proof that the Iwaniec-Sarnak mollifier is optimal in a wide class of one-piece mollifiers.