Nevanlinna’s omitted-values and Blaschke-divisor question

Nevanlinna’s question asks whether a meromorphic function on the whole complex plane that omits three values in a half-plane must have bounded type there. The question is described as originating with Nevanlinna and remaining unsettled.

Known results

• Nevanlinna proved a positive answer for meromorphic functions of finite order.
• Ostrovskii weakened this to $\int_1^\infty \frac{\log^+T(r;F)}{r^2}\,dr<\infty$.
• The modular function omits $0$ and $1$ in the upper half-plane but is not of bounded type there; it does not extend to resolve the entire-plane problem.

April 7, 2026 claimed counterexample

The preprint A Counterexample to Nevanlinna’s Century-Old Half-Plane Problem claims a nonconstant meromorphic $F\colon\mathbb{C}\to\widehat{\mathbb{C}}$ with

$$F^{-1}\{0,1,\infty\}\subset\mathbb{R},$$

while $F|_{\mathbb{H}}\notin N(\mathbb{H})$. This would give a negative answer. A related daily-news report describes the same claimed construction; neither source supplies independent verification.

Current status (as of August 2026): The original question remains mathematically unverified, with a preprint claiming a complete counterexample but no independent confirmation recorded.