The implication fails in the constructed case
The preprint addresses a precise question in complex analysis: if a meromorphic function on the complex plane omits three distinct spherical values in the upper half-plane, must it be of bounded type there? Its construction gives a nonconstant meromorphic extension F on the complex plane. The associated upper-half-plane function f omits 0, 1 and infinity, but the proof concludes that f is not in the Nevanlinna class, the bounded-type condition named in the question.
The question contains two separate pieces. The omission condition concerns values f does not take in the upper half-plane; the proposed conclusion concerns membership in the Nevanlinna class. Here omission holds for the three displayed values while membership fails, so the construction answers the implication with a counterexample. It does not, by itself, say that every function omitting three values has the same behavior.
A domain assembled from Farey edges
To build the example, the construction uses H, the upper half-plane, and a simply connected subdomain bounded by a locally finite chain of Farey edges. It also uses a suitably normalized Riemann map, a conformal map that transfers the setup to that domain. This provides the geometric framework for the function and for the Green-function estimates that follow.
The boundary is assembled through selected bays. Each bay is given a finite Farey refinement that meets the required local Green-energy inequality. The refinements are then part of the construction of the final domain, where the corresponding Green-weighted spherical-area quantity is examined.
The analytic signal comes from the modular lambda function
Alongside this geometry, the analysis examines the modular lambda function near the real axis. For sufficiently small positive heights, a period integral involving the positive part of the logarithm of its modulus has a lower bound that grows logarithmically in reciprocal height. The estimate describes the function's behavior as the boundary is approached.
The paper also proves the stated divergence of a height-weighted spherical-area integral for the modular lambda function. The result holds for every real horizontal offset and every positive cutoff height, in each period strip specified by the construction.
That area calculation is paired with a Green-weighted spherical-area integral on H. In the proof, an infinite value is used as a sufficient obstruction to membership in N(H), the notation for the Nevanlinna class. The criterion is one-way: it is enough to rule out membership when the integral diverges, but it is not offered as a complete characterization of all nonmembership.
From local geometry to a global function
The final Farey-boundary domain satisfies the paper's infinite Green-weighted spherical-area conclusion. Separately, the proof concludes that the conformally transferred function f is outside N(H). These are the domain-level and function-level statements at the center of the counterexample.
The construction extends beyond H. Its normalized map fixes i and sends infinity to infinity. Schwarz reflection extends the composition of the modular lambda function with that map meromorphically to the complex plane, and every preimage of 0, 1 and infinity is real.
That extension is the nonconstant meromorphic function F used in the construction, while f is its upper-half-plane function. The result combines a global extension, omission of 0, 1 and infinity in H, and failure of Nevanlinna-class membership for f.
What the example leaves open
The authors also state that F has infinite order. They interpret this as placing the example outside the scope of the finite-order theorem. The paper further reports that the deficiencies at 0, 1 and infinity are all zero.
Those properties belong to this particular construction. The work provides one deliberately built function and domain; it does not characterize all meromorphic functions that omit three values, nor does it establish that all such functions fall outside the Nevanlinna class.
The paper leaves the complete singular-value set unresolved. It does not establish that the set is exactly 0, 1 and infinity, because possible asymptotic values at infinity require separate analysis. That open point is distinct from the narrower result that the constructed restriction is outside N(H).
Preprint status and disclosure
The supplied document is an arXiv version v1 preprint dated 25 August 2026. Its acknowledgements report Teng Zhang's institutional and program support, including grant xzy022024045. They also disclose AI-assisted exploration under the authors' mathematical supervision and guidance and state that the authors take full responsibility for the manuscript.
Paper data and sources
Original title: A counterexample to Nevanlinna's century-old half-plane problem
Authors: Yixin He, Teng Zhang
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-25
DOI: Not available
Original paper · Full text