Contractivity Abscissas of Dirichlet Partial Sums

Definitions and mathematical statement

For $\sigma\geq0$, let \[ \mathbb C_\sigma=\{s\in{\mathbb{C}}:\operatorname{Re}s>\sigma\}. \] Let $H^\infty(\mathbb C_\sigma)$ denote the space of bounded holomorphic functions on $\mathbb C_\sigma$, with norm \[ \|g\|_{H^\infty(\mathbb C_\sigma)}=\sup_{s\in\mathbb C_\sigma}|g(s)|. \] Let $\mathcal D_\infty$ consist of those $f\in H^\infty(\mathbb C_0)$ represented in some right half-plane by an ordinary Dirichlet series \[ f(s)=\sum_{n=1}^{\infty}a_n n^{-s},\qquad a_n\in{\mathbb{C}}, \] where $n^{-s}=\exp(-s\log n)$. This makes explicit the Dirichlet-series interpretation of the source's notation $f(s)=\sum_{n\geq1}a_nn^{-s}\in H^\infty(\mathbb C_0)$. For each positive integer $N$, define the partial sum \[ S_N(f)(s)=\sum_{n=1}^{N}a_n n^{-s}. \] Define \[ \sigma_N(f)=\inf\left\{\sigma>0: \|S_N(f)\|_{H^\infty(\mathbb C_\sigma)} \leq\|f\|_{H^\infty(\mathbb C_\sigma)}\right\}, \qquad \sigma_N=\sup_{f\in\mathcal D_\infty}\sigma_N(f). \] For definiteness, these quantities are interpreted in $[0,+\infty]$, with $\inf\varnothing=+\infty$. The supplied record does not specify this convention or assert that the sets defining the infima are nonempty. Both norms above are taken on the same shifted half-plane, exactly as in the supplied statement; no normalization of $f$ is imposed.

The question attributed to Bayart in the supplied curation is whether \[ \lim_{N\to\infty}\sigma_N=0. \] Equivalently, is it true that for every $\varepsilon>0$ there exists a positive integer $N_0$ such that, for every integer $N\geq N_0$ and every $f\in\mathcal D_\infty$, one has $\sigma_N(f)<\varepsilon$? This asks about the infima as defined and does not assume that the defining norm inequality is monotone in $\sigma$.

Short English statement

For each Dirichlet partial sum, measure the infimum of positive half-plane shifts on which its supremum norm does not exceed that of the original bounded Dirichlet series. Does the supremum of these infima over all such series tend to zero as the number of terms tends to infinity? Both norms use the same shifted half-plane. The supplied curation labels this question open; that status has not been independently verified.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $30003647$, title Contractivity Abscissas of Dirichlet Partial Sums, exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection.
• 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. https://doi.org/10.4171/owr/2017/51. The supplied background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature reference: Oberwolfach Report 51, Bayart problem. https://doi.org/10.4171/owr/2017/51. The supplied literature assessment, dated exactly 2026-08-21, states that Bayart's question remains open in the available literature. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment or source visit.
• Attribution is to Bayart in the supplied background and literature assessment; the structured proposer field is empty. The supplied proposed year is $2017$, whereas the source citation labels the workshop $2018$.