Does there exist an absolute constant C>0 such that every Dirichlet polynomial f(s)=∑_n=1^Na_n n^-s satisfies [ int_0^1|f(frac12+sigma)|,d sigma≤ C lim_T→∞frac12Tint_-T^T|f(it)|,dt? ]

Fejér–Riesz inequality problem for Dirichlet series

The problem concerns a boundary-to-interior inequality for Dirichlet series and its equivalent-looking coefficient estimate in Hardy spaces, with implications for the bounded-symbol question associated with the multiplicative Hilbert matrix. Earlier literature treated that bounded-symbol question as open.

September 2026 preprint

The preprint A Fejér--Riesz inequality for Dirichlet series states that it proves the requested inequality and derives the associated Hardy-space coefficient estimate. Its abstract also describes proving a conjecture of Brevig, Ortega-Cerdà, Seip, and Zhao connected with the Riesz projection. The claim is mathematically relevant but remains unrefereed.

Current status (as of September 2026): The inequality and associated coefficient estimate are claimed proved in an unrefereed arXiv preprint, but independent verification is not recorded.

Fejér–Riesz inequality problem for Dirichlet series

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