On optimality of mollifiers

TL;DR

This paper investigates the optimal construction of mollifiers used in number theory, providing criteria for their optimality and demonstrating the optimality of specific mollifiers in the context of Dirichlet L-functions.

Contribution

It introduces criteria for identifying optimal mollifiers based on mollified moments and proves the optimality of Michel-Vanderkam and Iwaniec-Sarnak mollifiers in their respective classes.

Findings

Michel-Vanderkam mollifier is optimal among balanced two-piece mollifiers.

Iwaniec-Sarnak mollifier is optimal among one-piece mollifiers.

Provides a new proof of the Iwaniec-Sarnak mollifier's optimality.

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.