Critical Mean Values of Dirichlet Polynomials

Definitions and mathematical statement

Let $T>1$ and let $N$ be a positive cutoff satisfying \[ T^{1/2}<N<T. \] Set \[ \kappa=\frac{\log T}{\log N}, \qquad N^{\kappa}=T, \qquad 1<\kappa<2. \] For coefficients $a_n$ with $|a_n|\leq 1$, define the Dirichlet polynomial \[ D(s)=\sum_{\substack{n\in{\mathbb{N}}\\1\leq n\leq N}}a_n n^{-s}, \qquad s\in{\mathbb{C}}, \qquad n^{-s}=\exp(-s\log n). \] The supplied question asks whether, for every $\varepsilon>0$, \[ \int_0^T |D(it)|^{2\kappa}\,{\,\mathrm{d}} t \ll T^{2+\varepsilon}. \] Under a fully uniform interpretation, this means asking whether for every $\varepsilon>0$ there exists $C_{\varepsilon}>0$ such that \[ \int_0^T |D(it)|^{2\kappa}\,{\,\mathrm{d}} t \leq C_{\varepsilon}T^{2+\varepsilon} \] for every admissible $T,N$ and every admissible coefficient sequence. The source does not explicitly specify whether the implied constant must be uniform as $\kappa$ varies in $(1,2)$, or whether $\kappa$ is fixed and a constant $C_{\varepsilon,\kappa}$ is allowed. This quantifier distinction remains unresolved. Complex coefficients are the usual interpretation used in this draft, but their field is not expressly specified in the record. Likewise, the record does not specify whether $N$ must be integral; the sum is meaningful for a real cutoff. These conventions require confirmation against the source.

Short English statement

The question asks whether a Dirichlet polynomial with coefficients of modulus at most one satisfies the displayed mean-value bound when $T^{1/2}<N<T$ and its moment exponent is determined by $N^\kappa=T$. Uniformity in $\kappa$, the coefficient field, and the cutoff convention require confirmation. The supplied record labels the problem open.

Sources

• Local UnsolvedMath registry: supplied complete record for permanent ID $30003642$, titled Critical Mean Values of Dirichlet Polynomials, exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2017.
• Supplied source citation: Mini-Workshop: Interplay between Number Theory and Analysis for Dirichlet Series (2018), Oberwolfach Reports, pp. 3035--3069, DOI 10.4171/owr/2017/51. Literature-source label: "Oberwolfach Report 51, Dirichlet-polynomial discussion." https://doi.org/10.4171/owr/2017/51. The supplied citation uses 2018, whereas the DOI and proposed-year metadata use 2017; their relationship has not been independently checked.
• Supplied literature source: "Oberwolfach Report 51, full text." https://ems.press/content/serial-article-files/46710.
• Literature assessment dated exactly 2026-08-21: supplied dated curation, not independently verified. It labels the bound open and unproved, and states that the Hardy--Littlewood majorant implication is unavailable because that majorant conjecture is false. The background describes this as dated literature triage derived from the supplied OWR curation and requests independent verification of time-sensitive status and cited claims.
• The registry supplies the statement-status label corrected_verified and reports a manual semantic, typographic, and LaTeX review. These are supplied editorial assertions, not independent verification of the formulation or mathematical correctness of this draft.