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Implications of subconvexity bounds for the moments of zeta

Kevin Smith

Abstract

It is well-known that upper bounds for moments of the Riemann zeta function $ζ(s)$ have implications for subconvexity bounds. In this paper we explore some implications in the opposite direction using functional analysis in the right-half of the critical strip. The main results characterise potential transitions in the behaviour of the moments.

Implications of subconvexity bounds for the moments of zeta

Abstract

It is well-known that upper bounds for moments of the Riemann zeta function have implications for subconvexity bounds. In this paper we explore some implications in the opposite direction using functional analysis in the right-half of the critical strip. The main results characterise potential transitions in the behaviour of the moments.
Paper Structure (14 sections, 9 theorems, 97 equations)

This paper contains 14 sections, 9 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Preliminaries. Almost-periodicity
  3. Proof of Theorems \ref{['zetacns']} and \ref{['thzeta']}
  4. Theorem \ref{['zetacns']}
  5. Theorem \ref{['thzeta']}
  6. Proof of Propositions \ref{['holder']} and \ref{['prop4']}
  7. Proposition \ref{['holder']}
  8. Proof of Proposition \ref{['prop4']}
  9. proofs of the auxiliary results
  10. Proposition \ref{['sig']}
  11. Proposition \ref{['lem1']}
  12. Lemma \ref{['mainlemma']}
  13. Lemma \ref{['ugroup']}
  14. Acknowledgement

Key Result

Proposition 1

The following statements are equivalent. If $f$ satisfies them, we say that $f$ is not concentrated on a null set (not CNS). Moreover, if $\int_{0}^T|f|^2dt\gg T$, we may include If also $\int_{0}^T|f|^2dt\ll T$ then condition (c) may be replaced with

Theorems & Definitions (10)

  • Proposition 1: Concentration on a null set
  • Theorem 1
  • Proposition 2
  • Theorem 2
  • Proposition 3
  • Proposition 4
  • Lemma 1
  • Lemma 2
  • Proposition 5
  • proof : Proof of Proposition \ref{['zo']}