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 $f(s) = \sum_{n\geq 1} a_n n<sup>{-s}$ belonging to the Hardy space $\mathscr{H}<sup>1$ 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.