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A note on zero density results implying large value estimates for Dirichlet polynomials

Kaisa Matomäki, Joni Teräväinen

Abstract

In this note we investigate connections between zero density estimates for the Riemann zeta function and large value estimates for Dirichlet polynomials. It is well known that estimates of the latter type imply estimates of the former type. Our goal is to show that there is an implication to the other direction as well, i.e. zero density estimates for the Riemann zeta function imply large value estimates for Dirichlet polynomials.

A note on zero density results implying large value estimates for Dirichlet polynomials

Abstract

In this note we investigate connections between zero density estimates for the Riemann zeta function and large value estimates for Dirichlet polynomials. It is well known that estimates of the latter type imply estimates of the former type. Our goal is to show that there is an implication to the other direction as well, i.e. zero density estimates for the Riemann zeta function imply large value estimates for Dirichlet polynomials.
Paper Structure (10 sections, 15 theorems, 81 equations)

This paper contains 10 sections, 15 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Outline of the proof of Theorem \ref{['th:density->large']}
  3. Auxiliary results
  4. Partial summation and proof of Lemma \ref{['le_density1']}
  5. From large Dirichlet polynomial values to large zeta values --- proof of Lemma \ref{['le_FromRtoLargeZeta']}
  6. From large zeta values to zeros of zeta --- proof of Lemma \ref{['le_density3']}
  7. From large zeta values to large Dirichlet polynomial values --- proof of Lemma \ref{['le_zetabig->Rbig']}
  8. The inductive argument --- proof of Lemma \ref{['le_density2']}
  9. Proof of Theorem \ref{['thm_DH']}
  10. Proof of Proposition \ref{['prop:converse']}

Key Result

Proposition 1.1

Let $\varepsilon >0, \nu \in (0, 1/2],$ and $T \geq 3$. Let Then we can partition $\mathcal{T} = \mathcal{T}_1 \cup \mathcal{T}_2$ in such a way that $\# \mathcal{T}_1 \ll_\varepsilon T^{2\nu+2\varepsilon}$ and, for each zeta zero $\rho = \beta + i\gamma \in \mathcal{T}_2$, there exists $M \in [T^\varepsilon, T^{1/2}/2]$ and $M' \in (M, 2M]$ such that

Theorems & Definitions (30)

  • Proposition 1.1
  • Theorem 1.2: Zero density results imply large value estimates
  • Remark 1.3
  • Conjecture 1.4: Stronger density hypothesis
  • Theorem 1.5
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3: Large value of $\zeta(s)$ implies the existence of a nearby zero
  • Lemma 2.4
  • Lemma 2.5
  • ...and 20 more