Intermediate BMO Spaces for Dirichlet Series

Definitions and mathematical statement

A Dirichlet series has the form $f(s)=\sum_{n=1}^{\infty}a_n n^{-s}$, with complex coefficients $a_n$ and complex variable $s$, on a domain where the series or its specified analytic continuation defines a function. Here BMO denotes bounded mean oscillation. The supplied record does not specify the candidate Dirichlet-series BMO spaces, their norms, or the ambient domain and meaning of $H^\infty$.

The recorded question asks whether there exists a BMO-type space $\mathcal B$ of Dirichlet-series functions, intermediate between $H^\infty$ and the candidate BMO spaces, satisfying both of the following requirements:

1. For every $f\in\mathcal B$, every real $\sigma>0$, and every exponent described in the source as $p<\infty$, \[ \limsup_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}|f(\sigma+it)|^p\,{\,\mathrm{d}} t<\infty. \]
2. For every real $\sigma>\tfrac12$, the corresponding horizontal shift of the alternating zeta function belongs to $\mathcal B$.

These requirements retain unresolved source qualifications. The lower bound and domain for $p$ are absent; in particular, the record does not distinguish $0<p<\infty$ from $1\le p<\infty$. The usual alternating zeta function is $\eta(s)=\sum_{n=1}^{\infty}(-1)^{n-1}n^{-s}$ for $\operatorname{Re}s>0$. Under the conventional right-translation notation $(S_\sigma g)(s)=g(s+\sigma)$, the second requirement would read $S_\sigma\eta\in\mathcal B$ for every $\sigma>\tfrac12$; the record does not explicitly define that translation convention. Likewise, interpreting "intermediate" as $H^\infty\subseteq\mathcal B\subseteq X$ requires identification of the candidate space $X$, and clarification of whether inclusions must be continuous or strict and whether every candidate is intended. The original statement refers to properties (2) and (3) of the report, whose full text is not supplied. Consequently, this is a qualified reconstruction of the recorded existence question, requiring human review before it can serve as a fully specified formulation.

Short English statement

The question asks for an intermediate BMO-type space of Dirichlet series whose functions have finite vertical mean $p$-moments on every line $\operatorname{Re}s=\sigma>0$, and which contains the specified shifts of the alternating zeta function for every $\sigma>\tfrac12$. The candidate spaces, exponent range, inclusion requirements, and shift convention remain insufficiently specified.

Sources

• Local registry: supplied UnsolvedMath source record, permanent ID $30002502$, title Intermediate BMO Spaces for Dirichlet Series, source problem code catalogue entry, Oberwolfach Reports Open Problems collection. The record gives proposed year 2014 and identifies no individual proposer.
• Supplied report citation: Dirichlet Series and Function Theory in Polydiscs (2014), Oberwolfach Reports, pp. 335--393, DOI 10.4171/owr/2014/06. https://doi.org/10.4171/owr/2014/06. This is also the supplied literature-source link labeled "OWR report." The original registry statement refers to properties (2) and (3) in this report.
• Supplied dated literature curation, checked 2026-08-21: the record labels the problem open, states that the report records it as open, and reports that the literature located did not identify a space satisfying both required properties. Its assessment calls these "multiplier properties." This is supplied dated curation, not an independently verified literature assessment.
• The registry labels the statement corrected_verified and reports checking it against the official report. That verification claim is preserved as source metadata; neither the report nor the claimed verification has been independently checked for this draft.