Problem 1115

../

claims/: The 3 claim pages of Problem 1115, one per claimant's result; the problem's standing derives from them.

---

Statement. Let $f(z)$ be an entire function of finite order, and let $\Gamma$ be a rectifiable path on which $f(z)\to \infty$. Let $\ell(r)$ be the length of $\Gamma$ in the disc $\lvert z\rvert<r$.

Find a path for which $\ell(r)$ grows as slowly as possible, and estimate $\ell(r)$ in terms of $M(r)=\max_{\lvert z\rvert=r}\lvert f(z)\rvert$.

In particular, can such a path $\Gamma$ be found for which $\ell(r)\ll r$?

Formulation. “Rectifiable” is understood locally, as the wording's own $\ell(r)$, a finite length inside each disc, presumes: a path on which $f(z)\to\infty$ tends to infinity, since an entire function is bounded on compact sets, so it cannot have finite total length. Gol'dberg and Eremenko (p. 509) state the conjecture and their theorems for any asymptotic curve, with $l(r,\Gamma)$ the length of its part in the disc, and the library's statements of their Theorems 1 and 2 read them for locally rectifiable paths. The quantity $\ell(r)$ is the length of all portions inside the disc, including any later returns, not merely the arc up to its first intersection with the circle. The definite question asks whether every entire function of finite order has such a path.

Status. SOLVED, the site's label, with commentary recording a disproof by Gol'dberg and Eremenko. Their Theorems 1 and 2 give entire functions of order zero, growing barely above Hayman's logarithmic-square threshold, and of every finite order, such that no path on which $f\to\infty$ has length $O(r)$ inside the disc of radius $r$; the answer to the problem's question is therefore no, and the standing is solved, disproved (their claim page). The standing records a disproof where the label says only SOLVED, because the problem's definite question asks whether every entire function of finite order has a path with $\ell(r)\ll r$, and the accepted full claims refute that. Toppila's 1980 note proves the same theorem independently (Toppila's claim page), and Hayman's 1960 theorem, the yes-instances for functions with $\log M(r,f)=O((\log r)^2)$, is a partial claim (its claim page). The wider request for an optimal length estimate in terms of $M(r)$ is not supplied by that counterexample, and no optimal replacement bound for every growth class is asserted here.

Source. erdosproblems.com/1115, accessed 2026-09-05. Cite as: T. F. Bloom, Erdős Problem #1115, https://www.erdosproblems.com/1115, accessed 2026-09-05. As of that date the site's discussion listed no comments, expositions or proof claims, and the page reported its last edit as 29 December 2025.

References.

• [GoEr79] Gol'dberg, A. A. and Eremenko, A. E., Asymptotic curves of entire functions of finite order. Mat. Sb. (N.S.) (1979), 555-581, 647. The compiled source is the English translation, On asymptotic curves of entire functions of finite order, Math. USSR-Sbornik 37 (1980), 509–533; see the source digest.
• [Ha60] W. Hayman, Defective values and asymptotic paths. Matematika (1960), 21-27.
• [Ha60b] Hayman, W. K., Slowly growing integral and subharmonic functions. Comment. Math. Helv. (1960), 75-84.
• [Ha74] Hayman, W. K., Research problems in function theory: new problems. (1974), 155-180.

Formalization. The site shows no formalized statement and formal-conjectures has no file for the problem. Boris Alexeev's repository holds Erdos1115.lean (added 17 August 2026), a Lean development that declares itself a formalization of Gol'dberg and Eremenko's solution, with Codex and GPT-5.6 Sol as its formal authors; it is linked from their claim page, and this corpus has not built it.

Current assessment

The search checked the author bibliography, publisher record, the existing survey, related primary-paper titles, and E1115 announcements on X. It found no replacement or correction to the counterexample theorem. No claim of an exhaustive current optimal upper bound is made from that search.

The finite-order assertions of Theorems 1 and 2 and the common spiral-barrier step are recorded on their source pages, the theorems with proof sketches. Theorem 4's distinct conformal-construction proof is a statement and sketch with an explicit gap list. The wider quantitative coverage and its uncompiled sources are described below.

Progress

The source's introduction attributes the conjecture to Hayman's 1960 lecture [Ha60]; Hayman's 1974 collection [Ha74], Problem 2.41, attributes the question to Erdős. The 1979 counterexample paper resolves the linear-length question negatively. Its English translation, published in 1980, is the version used below.

Hayman's positive result [Ha60b], as stated on p. 509 of [GoEr79], is that a nonconstant entire function satisfying

$$\log M(r,f)=O((\log r)^2)$$

tends to infinity on rays of almost every argument. One may therefore choose $\ell(r)=r$. The result has its own claim page, a partial claim covering that growth class; its original proof is not rewritten on the library's result pages.

Known Results

Theorem 1 of Gol'dberg–Eremenko shows that for every function $\phi(r)\to\infty$ there is an entire function of order zero such that

$$\log M(r,f)=O(\phi(r)(\log r)^2),$$

yet every locally rectifiable asymptotic path to infinity satisfies

$$\limsup_{r\to\infty}\frac{\ell(r)}r=\infty.$$

Thus no unbounded multiplicative relaxation of Hayman's logarithmic-square growth hypothesis guarantees a linear-length path. The proof uses small-value spiral barriers built by polynomial approximation, then an infinite product with controlled zero counting.

Theorem 2 constructs such counterexamples of every prescribed finite order $\rho\ge0$. These finite-order assertions are recorded with proof sketches on their source pages, and the common spiral-barrier argument has its own source page.

For the finite-asymptotic-value variant in Hayman's Problem 2.41, Theorem 4 gives a function of every order $\rho\ge1/2$ having $0$ as an asymptotic value but no linear-length path to $0$. For finite $\rho$ its lower order equals its order. That distinct conformal-construction proof is a statement and sketch with an explicit gap list. The order-$1/2$ assertion does not impose normal type.

Wider quantitative question and literature coverage

The Hayman–Lingham survey, arXiv:1809.07200v2 (2018), Update 2.41, printed p. 38, records the Gol'dberg–Eremenko resolution. Update 2.7, p. 25, also cites Toppila's independent proof (1980), Ann. Acad. Sci. Fenn. Ser. A I Math. 5 (1980), 13–15. That note has its own claim page: it poses the question from Hayman's Problem 2.41, proves the order-zero counterexample for every increasing $\phi(r)\to\infty$, and acknowledges Gol'dberg and Eremenko's priority in a closing remark.

For positive upper bounds, Update 2.7 cites K. H. Chang, Asymptotic values of entire and meromorphic functions, Sci. Sinica 20 (1977), 720–739, for a bound $O(r^{1+\rho/2+\varepsilon})$ for each $\varepsilon>0$. In that update the length is defined only up to the first intersection with $|z|=r$. This reported bound is not being asserted here for the total in-disc length in the present question; Chang's paper and its length metric are not compiled here.

Update 2.57, p. 44, points to Anderson's smooth-growth theorem (1979), which provides nearly radial asymptotic paths to a deficient value under the extra condition $T(2r,f)\sim T(r,f)$ on the Nevanlinna characteristic. Anderson's proof and the later work it led to are not compiled here. These qualifications concern quantitative coverage, not the established disproof of the linear-length conjecture.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• goldberg_1979_asymptotic_curves_entire_functions_finite_order
• goldberg_1979_asymptotic_curves_entire_functions_finite_order / spiral_barriers
• goldberg_1979_asymptotic_curves_entire_functions_finite_order / theorem_1
• goldberg_1979_asymptotic_curves_entire_functions_finite_order / theorem_2
• goldberg_1979_asymptotic_curves_entire_functions_finite_order / theorem_4
• goldberg_1979_asymptotic_curves_entire_functions_finite_order / theorem_5
• hayman_lingham_2018_research_problems_function_theory
• hayman_lingham_2018_research_problems_function_theory / problem_2_41
• hayman_lingham_2018_research_problems_function_theory / problem_2_7