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

Abstract: Hilbert's inequality states that $$ \left\vert\sum_{m=1}^N\sum_{n=1\atop n\neq m}^N\frac{z_m\overline{z_n}}{m-n}\right\vert\le C_0\sum_{n=1}^N\left\vert z_n\right\vert^2, $$ where $C_0$ is an absolute constant. In 1911, Schur showed that the optimal value of $C_0$ is $\pi$.

In 1974, Montgomery and Vaughan proved a weighted generalization of Hilbert's inequality and used it to estimate mean values of Dirichlet series. This generalized Hilbert inequality is important in the theory of the large sieve. The optimal constant $C$ in this inequality is known to satisfy $\pi\le C<\pi+1$. It is widely conjectured that $C=\pi$. In this talk, I will describe the known approaches to obtain an upper bound for $C$, which proceed via a special case of a parametric family of inequalities. We analyze the optimal constants in this family of inequalities. A corollary is that the method in its current form cannot imply an upper bound for $C$ below $3.19$.

number theory

Audience: researchers in the discipline

Combinatorial and additive number theory (CANT 2022)