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Applying the Resonance Method to $\textrm{Re}\left(e^{-iθ}\logζ(σ+it)\right)$

Mikko Jaskari

Abstract

We apply the resonance method to Montgomery's convolution formula for $\textrm{Re}\left(e^{-iθ}\logζ(σ+it)\right)$ in the strip $1/2 < σ< 1$. This gives new insight into maximal values of $\textrm{Re}\left(e^{-iθ}\logζ(σ+it)\right)$ for $t \in [T^β,T]$ for all $β\in (0,1)$ and real $θ$.

Applying the Resonance Method to $\textrm{Re}\left(e^{-iθ}\logζ(σ+it)\right)$

Abstract

We apply the resonance method to Montgomery's convolution formula for in the strip . This gives new insight into maximal values of for for all and real .
Paper Structure (3 sections, 11 theorems, 70 equations)

This paper contains 3 sections, 11 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. The Convolution Formula
  3. Proof of Theorem \ref{['th1.1']} by the Resonance Method

Key Result

Theorem 1.1

Fix $\sigma \in (1/2,1)$, $\beta \in (0,1)$ and ${0 < \kappa < \min(\sigma-1/2,1-\beta)}$. Then there exists a function $\upsilon : (1/2,1) \to \mathbb{R}_+$ which satisfies where $c$ is a positive constant independent from any of the chosen parameters $\sigma, \beta$ or $\kappa$ such that for any $\theta \in \mathbb{R}$ and sufficiently large $T$ we have

Theorems & Definitions (25)

  • Theorem 1.1
  • Corollary 1.2
  • Lemma 2.1
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Definition 3.5
  • Definition 3.6
  • Definition 3.7
  • ...and 15 more