Year: 2026
Type: preprint
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 $f(s) = \sum_{n\geq 1} a_n n^{-s}$ 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.
Source: arXiv (Cornell University)
Author: Perfekt, Karl-Mikael
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Norges Forskningsråd
Award: G3293145243
OpenAlex ID: W7208719547