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On the Montgomery--Vaughan weighted generalization of Hilbert's inequality

Wijit Yangjit

Abstract

This paper concerns the problem of determining the optimal constant in the Montgomery--Vaughan weighted generalization of Hilbert's inequality. We consider an approach pursued by previous authors via a parametric family of inequalities. We obtain upper and lower bounds for the constants in inequalities in this family. A lower bound indicates that the method in its current form cannot achieve any value below $3.19497$, so cannot achieve the conjectured constant $π$. The problem of determining the optimal constant remains open.

On the Montgomery--Vaughan weighted generalization of Hilbert's inequality

Abstract

This paper concerns the problem of determining the optimal constant in the Montgomery--Vaughan weighted generalization of Hilbert's inequality. We consider an approach pursued by previous authors via a parametric family of inequalities. We obtain upper and lower bounds for the constants in inequalities in this family. A lower bound indicates that the method in its current form cannot achieve any value below , so cannot achieve the conjectured constant . The problem of determining the optimal constant remains open.
Paper Structure (13 sections, 17 theorems, 79 equations, 1 figure)

This paper contains 13 sections, 17 theorems, 79 equations, 1 figure.

Table of Contents

  1. Introduction
  2. History of the problem
  3. Main results
  4. Preliminaries
  5. Eigenvalues of generalized weighted Hilbert matrices
  6. A weighted spacing lemma and Shan's method
  7. Proofs of Theorems \ref{['C1C1/2inequality']} and \ref{["Preissmann'sTheorem"]}
  8. Proof of Theorem \ref{['C1C1/2inequality']}
  9. Proof of Theorem \ref{["Preissmann'sTheorem"]}
  10. Proofs of Theorems \ref{['boundedness']} and \ref{['C(1)exact']}
  11. Proof of Theorem \ref{['boundedness']}
  12. Proof of Theorem \ref{['C(1)exact']}
  13. Proof of Theorem \ref{['main-negative']}

Key Result

Theorem 1.1

We have $\overline{C}_1\le\sqrt{\frac{\pi^2}{3}+2\overline{C}\left(\frac{1}{2}\right)}$.

Figures (1)

  • Figure 1: The plot of $G_K\left(\frac{x}{K+1}\right)$ for $K=1,\dots,25$ and $0<x<1$.

Theorems & Definitions (34)

  • Theorem 1.1
  • Theorem 1.2: Preissmann
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • ...and 24 more