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 3.19497 3.19497 , so cannot achieve the conjectured constant π \pi . The problem of determining the optimal constant remains open.