Zero-Free Dirichlet Polynomials on the Unit Line

Definitions and mathematical statement

For $s\in{\mathbb{C}}$, define the Dirichlet polynomial \[ D(s)=1+2^{-s}+3^{-s}+5^{-s},\qquad n^{-s}=\exp(-s\log n), \] where $\log n$ is the real natural logarithm for $n\in\{2,3,5\}$. The nonvanishing question is whether \[ \forall s\in{\mathbb{C}},\qquad \operatorname{Re}(s)=1\ \Longrightarrow\ D(s)\ne0. \] Equivalently, writing $s=1+it$, it asks whether \[ \forall t\in{\mathbb{R}},\qquad 1+\frac{e^{-it\log 2}}{2}+\frac{e^{-it\log 3}}{3}+\frac{e^{-it\log 5}}{5}\ne0. \] The original source record also presents the following as an equivalent Tauberian formulation. Is it true that every function $f:(0,\infty)\to(0,\infty)$ that is nondecreasing, meaning \[ 0<x\le y\ \Longrightarrow\ f(x)\le f(y), \] and satisfies \[ \lim_{x\to\infty}\frac{f(x)+f(x/2)+f(x/3)+f(x/5)}{x}=\frac{61}{30} \] necessarily satisfies \[ \lim_{x\to\infty}\frac{f(x)}{x}=1? \] These limits express the supplied formulas $f(x)+f(x/2)+f(x/3)+f(x/5)=(61/30+o(1))x$ and $f(x)=(1+o(1))x$, with the asymptotic direction interpreted as $x\to\infty$. The original record does not explicitly state that direction. The equivalence of the two questions is reported by the supplied record and is not established in this statement-only draft. No continuity or differentiability assumption on $f$ is supplied.

Short English statement

The problem asks whether $1+2^{-s}+3^{-s}+5^{-s}$ is nonzero at every complex point on the line $\operatorname{Re}(s)=1$. The original record also asks, as a stated equivalent, whether a positive nondecreasing function whose four indicated dilates have sum asymptotic to $(61/30)x$ must itself be asymptotic to $x$. The supplied open status and verification labels are curation claims requiring independent review.

Sources

• Local UnsolvedMath registry, supplied complete record for permanent ID $30003644$, source problem code catalogue entry, title "Zero-Free Dirichlet Polynomials on the Unit Line." Registry collection: data/unsolvedmath/problems.json; supplementary existing-record catalog: catalog/problems.json. This draft uses the supplied record without independently inspecting those files.
• Supplied primary-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 record attributes the problem to the workshop's open-problem discussions, gives proposed year 2017, and names no individual proposer.
• Supplied literature reference: Erdős--Ingham, Arithmetical Tauberian theorems. https://eudml.org/doc/207489.
• Supplied literature reference: Erdős Problems, problem 967. https://www.erdosproblems.com/latex/967.
• Supplied literature reference: Oberwolfach Report 51, open-problems session. https://doi.org/10.4171/owr/2017/51.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and describes this polynomial as the simplest unresolved case of the Erdős--Ingham nonvanishing question on $\operatorname{Re}(s)=1$. This is supplied dated curation, not independently verified. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry supplies statement status corrected_verified and states that the clean statement was checked against the official Oberwolfach report and preserves its hypotheses, notation, and open mathematical target. This is a supplied source-verification claim; no source was independently visited for this draft.