A Dirichlet Series with Exactly One Zero

Definitions and mathematical statement

Under the usual complex-coefficient convention, an ordinary Dirichlet series is an expression \[ F(s)=\sum_{n=1}^{\infty}a_n n^{-s},\qquad a_n\in{\mathbb{C}},\quad s\in{\mathbb{C}},\quad n^{-s}=\exp(-s\log n). \] For a real number $\sigma$, write \[ H_\sigma=\{s\in{\mathbb{C}}:\operatorname{Re}s>\sigma\}. \] Convergence in this half-plane means that the partial sums converge at every point of $H_\sigma$; their limit defines a holomorphic function there.

The supplied clean question, interpreted as counting distinct zeros, asks whether there exist $\sigma\in{\mathbb{R}}$, a sequence $(a_n)_{n\geq1}$ of complex numbers, and $s_0\in H_\sigma$ such that the series converges throughout $H_\sigma$ and \[ \{s\in H_\sigma:F(s)=0\}=\{s_0\}. \] The boundary line $\operatorname{Re}s=\sigma$ is excluded, and this formulation imposes no condition on zeros outside $H_\sigma$.

Unresolved qualifications. The supplied record does not explicitly specify the coefficient field; complex coefficients above express the usual convention, pending source review. It also does not specify whether zeros are counted with multiplicity. If exactly one zero is intended with multiplicity counted, the displayed condition must additionally require $F'(s_0)\ne0$. The wording does not explicitly require $H_\sigma$ to be the maximal open half-plane of convergence; whether that stronger requirement was intended needs verification.

The original statement also asks whether $D(\lambda)$ is continuous and strictly decreasing as a function of $\lambda$. Neither $D$ nor the domain of $\lambda$ is defined in the supplied record, so this additional question cannot be formulated precisely from the supplied evidence and remains unresolved here.

Short English statement

The main question asks whether an ordinary Dirichlet series can converge throughout an open right half-plane and vanish at exactly one point there. The coefficient convention, treatment of zero multiplicity, and intended half-plane require review. An additional original question about $D(\lambda)$ lacks its definition and parameter domain.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30002507$, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The supplied record gives proposed year 2014 and names 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. Literature-source label: Oberwolfach Report 06/2014. https://doi.org/10.4171/owr/2014/06.
• Supplied literature source: Hilberdink--Saias preprint. https://arxiv.org/abs/1812.11880.
• Supplied literature source: Current retraction discussion. https://mathoverflow.net/questions/330714/dirichlet-series-with-a-single-zero.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and reports that a proposed construction appeared as an arXiv preprint, but the associated paper was subsequently retracted, leaving the existence question unresolved according to that assessment. This is supplied dated curation, not an independently verified assessment of the literature or retraction claims.
• The registry labels the clean statement corrected_verified and reports that it was checked against the official Oberwolfach report. These are supplied verification claims; no independent source consultation or mathematical verification is asserted in this draft.