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On optimality of mollifiers

Martin Čech, Kaisa Matomäki

TL;DR

This work develops a general, quantitative framework for assessing mollifier optimality in the study of non-vanishing of central Dirichlet L-values. By formalizing mollified moments and introducing an efficiency metric, the authors derive criteria for when adding a new mollifier piece or forming a combined mollifier $M+\\alpha N$ improves the non-vanishing proportion, contingent on computable mollified moments. They prove the Michel–Vanderkam mollifier is optimal among balanced two-piece mollifiers and provide a new, self-contained argument for the Iwaniec–Sarnak mollifier’s optimality within a broad one-piece class, with averaging over moduli $q$ (and a weight function) crucial to the analysis. In the unbalanced case, their results indicate potential gains from additional mollifier components under moment computations, clarifying the limits of Pratt-type constructions. The methods hinge on a rigorous moment-calculus framework, inner-product techniques, and auxiliary estimates for Möbius sums, mean-value results, and incomplete Kloosterman sums, enabling precise comparisons across mollifier families and guiding future mollifier design for non-vanishing problems.

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.

On optimality of mollifiers

TL;DR

This work develops a general, quantitative framework for assessing mollifier optimality in the study of non-vanishing of central Dirichlet L-values. By formalizing mollified moments and introducing an efficiency metric, the authors derive criteria for when adding a new mollifier piece or forming a combined mollifier improves the non-vanishing proportion, contingent on computable mollified moments. They prove the Michel–Vanderkam mollifier is optimal among balanced two-piece mollifiers and provide a new, self-contained argument for the Iwaniec–Sarnak mollifier’s optimality within a broad one-piece class, with averaging over moduli (and a weight function) crucial to the analysis. In the unbalanced case, their results indicate potential gains from additional mollifier components under moment computations, clarifying the limits of Pratt-type constructions. The methods hinge on a rigorous moment-calculus framework, inner-product techniques, and auxiliary estimates for Möbius sums, mean-value results, and incomplete Kloosterman sums, enabling precise comparisons across mollifier families and guiding future mollifier design for non-vanishing problems.

Abstract

Mollifiers are used in a variety of contexts, for instance to study the non-vanishing of -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 -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.
Paper Structure (16 sections, 27 theorems, 243 equations)

This paper contains 16 sections, 27 theorems, 243 equations.

Table of Contents

  1. Introduction
  2. General results
  3. Proofs of the general results
  4. Reducing the proof of Theorem \ref{['thm:ISOptimality']} to computing mollified moments
  5. Reducing the proof of Theorem \ref{['thm:main general result DirL']} to computing mollified moments
  6. Auxiliary results
  7. Orthogonality of even characters and root numbers
  8. Sums with the Möbius function
  9. A mean value result
  10. A bound for sums of incomplete Kloosterman sums
  11. Proof of Proposition \ref{['prop:Psi_N_1']}
  12. Proof of Propositions \ref{['prop:PsiMN_1IS']} and \ref{['prop:PsiM1,N1']}
  13. Proof of Proposition \ref{['prop:PsiM_2N_1']}
  14. Main terms
  15. Error terms
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $\theta < 1/2$, let $q \geq 3$, and let be the Iwaniec-Sarnak mollifier. Let be a general mollifier of the same type, where $x_b$ are arbitrary complex coefficients normalized so that $x_1 = 1$. Assume that $x_b \ll q^\varepsilon$ for every $\varepsilon > 0$ and $b \in \mathbb{N}$. Then

Theorems & Definitions (57)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Example 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Corollary 2.6
  • Corollary 2.7
  • ...and 47 more