Breakthrough Tracker record

A preprint gives a counterexample to Nevanlinna’s century-old half-plane problem

Yixin He and Teng Zhang construct a nonconstant meromorphic function on the complex plane whose zeros, one-points and poles all lie on the real axis but whose restriction to the upper half-plane is not of bounded type. This would answer Nevanlinna’s unrestricted 1925 question negatively.

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Stable ID: math-nevanlinna-half-plane-counterexample-2026
Revision: math-nevanlinna-half-plane-counterexample-2026.v1
Field: Mathematics · Complex analysis and value-distribution theory
Evidence: Tier 1 · Peer reviewed: No
Record state: Provisional · Provisional claimed counterexample
Last checked: 2026-09-01

AI role

The authors state that AI-assisted exploration helped obtain the counterexample under their mathematical supervision. They do not name a model and accept responsibility for the final arguments.

Record details

Problem or result: Nevanlinna’s 1925 half-plane bounded-type problem for entire-plane meromorphic functions
Authors: Yixin He and Teng Zhang
Institutions: Fudan University; Xi’an Jiaotong University
Result date: First-version preprint submitted August 25, 2026

Why it matters

The problem had survived a century of positive results under finite-order and related growth conditions. The construction would show that those hypotheses cannot be removed.

Limits

The manuscript is not peer reviewed and no independent proof or specialist assessment was located. The counterexample concerns the unrestricted problem and does not overturn Nevanlinna’s finite-order theorem or later positive results under explicit growth conditions.

Sources

Primary: He and Zhang, half-plane counterexample manuscript
Primary context: Eremenko, Kulikov and Sodin, sharp bounded-type criteria

Correction and revision history

2026-09-01 — Added after primary-source, scope, status, AI-role and limitation review.

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