Research Problems in Function Theory — Problem 7.78

Definitions and mathematical statement

Let $(z_n)_{n\geq1}$ be pairwise distinct nonzero complex numbers with \[ \sum_{n=1}^{\infty}\frac1{|z_n|}<\infty. \] Then $|z_n|\to\infty$, and the series \[ F(z)=\sum_{n=1}^{\infty}\frac1{z-z_n} \] converges locally uniformly away from the $z_n$, defining a meromorphic function with a simple pole of residue one at each $z_n$.

L. A. Rubel's existence question is whether such a sequence can make $F$ zero-free: \[ \exists(z_n)_{n\geq1}\text{ as above}\ \forall z\in{\mathbb{C}}\setminus\{z_n:n\geq1\}, \qquad F(z)\ne0. \] The source says "for all $z\in{\mathbb{C}}$," although the displayed series is undefined at its poles. The precise interpretation above either excludes the poles or equivalently views $F$ meromorphically, with poles not counted as zeros; this domain qualification must be confirmed rather than hidden.

The source interprets $F$ as the field of unit-charged wires perpendicular to the plane and asks whether an equilibrium must always exist. Its formal logarithmic potential $\sum_n\log|z-z_n|$ needs additive renormalization: those summands do not tend to zero. A well-defined normalized potential away from the charges is \[ U(z)=\sum_{n=1}^{\infty}\log|1-z/z_n|, \qquad \partial_xU-i\partial_yU=F(z). \] Thus a field equilibrium away from charges is exactly a zero of $F$, with the complex-gradient convention explicit. This normalization explains the physical statement without asserting convergence of the raw potential.

The source also proposes an analogous question in ${\mathbb{R}}^n$ for $n\geq3$, but supplies no corresponding potential, convergence condition or precise higher-dimensional formulation; that suggested extension remains unspecified.

Short English statement

Can a summably sparse infinite set of equal plane charges produce a field with no equilibrium away from the charges? The field series is well defined under the stated reciprocal-distance sum; the original all-plane wording needs a pole convention and the raw logarithmic potential needs normalization. The supplied triage reports the question open in Hayman's 2018 update and no definitive resolution located; no new example or current-status assessment is claimed.

Sources

• UnsolvedMath permanent ID 2307078, source code catalogue entry, "Research Problems in Function Theory --- Problem 7.78." Registry curation and metadata: Ulam AI and UnsolvedMath Contributors, CC BY 4.0; updated 2026-08-06.
• Hayman, Research Problems in Function Theory (2018 source update), Problem 7.78, attributed to L. A. Rubel: https://arxiv.org/abs/1809.07200. Supplied source-TeX extraction accessed 2026-07-29. The metadata retains the update for status triage and marks status/release and rights review NEEDS_REVIEW.
• No additional authoritative result or reference is supplied for a resolution or the higher-dimensional suggestion. No missing citation is invented.