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

TL;DR

This paper investigates the best possible constant in a weighted generalization of Hilbert's inequality, providing bounds that show current methods cannot reach the conjectured optimal value, leaving the problem open.

Contribution

It offers new upper and lower bounds for the constants in the inequality's parametric family, highlighting limitations of existing approaches.

Findings

Lower bound of 3.19497 for the constant

Current methods cannot reach the conjectured value of π

The optimal constant problem remains unsolved

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 $\pi$. The problem of determining the optimal constant remains open.