Absolute-coefficient Hardy inequality for Dirichlet series

Prove the absolute-coefficient Hardy inequality \(|a_1|+\sum_{n\ge2}|a_n|/(\sqrt n\log n)\lesssim\|f\|_{\mathscr H^1}\) for every Dirichlet series \(f(s)=\sum_{n\ge1}a_n n^{-s}\) in \(\mathscr H^1\).

Background

The paper proves a signed Hardy inequality controlling ∣a1+∑n≥2an/(nlog⁡n)∣\left|a_1+\sum_{n\ge2}a_n/(\sqrt n\log n)\right| by the H1\mathscr H^1-norm, thereby establishing boundedness of a symbol for the multiplicative Hilbert matrix. It explicitly distinguishes that result from the stronger inequality involving the sum of the absolute values of all coefficients. The latter remains unresolved, and the author notes that the usual one-variable factorization argument does not directly transfer to Hardy spaces of Dirichlet series.

References

Before embarking on the proof of Theorem~\ref{thm:main}, let us note that the validity of the Hardy inequality

|a_1|+\sum_{n\ge2}\frac{|a_n|}{\sqrt n\log n} \lesssim|f|_{1}

remains an open problem.

— A Fejér--Riesz inequality for Dirichlet series  (2609.03855 - Perfekt, 3 Sep 2026) in Section 1, Introduction, immediately before Section 2