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arXiv · 2609.03855

A Fej\'er--Riesz inequality for Dirichlet series

Karl-Mikael Perfekt
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Abstract

We prove the following inequality for Dirichlet polynomials: \[ \int_0^1 |f(1/2+\sigma)|\,d\sigma\lesssim \lim_{T\to\infty} \frac{1}{2T} \int_{-T}^T |f(it)| \, dt. \] In particular, for a Dirichlet series $f(s) = \sum_{n\geq 1} a_n n^{-s}$ belonging to the Hardy space $\mathscr{H}^1$ 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.

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Karl-Mikael Perfekt. 2026-09-03. A Fej\'er--Riesz inequality for Dirichlet series. https://arxiv.org/abs/2609.03855

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