A Fejér--Riesz inequality for Dirichlet series

Abstract

We prove the following inequality for Dirichlet polynomials: \[ \int_0^1 |f(1/2+σ)|\,dσ\lesssim \lim_{T\to\infty} \frac{1}{2T} \int_{-T}^T |f(it)| \, dt. \] In particular, for a Dirichlet series $f(s)={\sum }_{n\ge 1}{a}_n{n}^{-s}$ belonging to the Hardy space ${\mathcal{H}}^1$ of Dirichlet series, \[ \left|a_1+\sum_{n=2}^\infty \frac{a_n}{\sqrt n\log n}\right| \lesssim \|f\|_{\mathscr{H}^1}. \] This answers a question raised previously in the literature and it proves that the multiplicative Hilbert matrix has a bounded symbol.

Improved exposition