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A Fejér--Riesz inequality for Dirichlet series
Abstract
We prove the following inequality for Dirichlet polynomials: ∫01 |f(1/2+σ)| dσ≲ limT→∞ 1/2T ∫−TT |f(it)| dt. In particular, for a Dirichlet series f(s) = ∑n≥ 1 an n−s belonging to the Hardy space ℋ1 of Dirichlet series, |a1+∑n=2∞ an/(√nlog n)| ≲ |f|ℋ1. This answers a question raised previously in the literature and it proves that the multiplicative Hilbert matrix has a bounded symbol.
Related
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BibTeX
@misc{perfekt2026fejérriesz,
title = {{A Fejér--Riesz inequality for Dirichlet series}},
author = {Karl-Mikael Perfekt},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.03855},
eprint = {2609.03855},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03855},
}
RIS
TY - UNPB
TI - A Fejér--Riesz inequality for Dirichlet series
AU - Karl-Mikael Perfekt
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.03855
UR - https://arxiv.org/abs/2609.03855
AB - We prove the following inequality for Dirichlet polynomials: ∫_0^1 |f(1/2+σ)| dσ≲ lim_T→∞ 1/2T ∫_−T^T |f(it)| dt. In particular, for a Dirichlet series f(s) = ∑_n≥ 1 a_n n^−s belonging to the Hardy space ℋ^1 of Dirichlet series, |a_1+∑_n=2^∞ a_n/(√nlog n)| ≲ |f|_ℋ^1. This answers a question raised previously in the literature and it proves that the multiplicative Hilbert matrix has a bounded symbol.
ER -