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Claim. The answer to the problem's question is no: an entire function $f$ of finite order need not have a locally rectifiable path on which $f\to\infty$ whose length $\ell(r)$ inside $|z|<r$ is $O(r)$. Theorem 1 (translation pp. 510--513): for every function $\phi(r)\to\infty$ there is an entire function $f$ of order zero with $\log M(r,f)=O(\phi(r)(\log r)^2)$ such that every locally rectifiable path on which $f\to\infty$ has $\limsup_{r\to\infty}\ell(r)/r=\infty$. Theorem 2 (pp. 513--516) gives such functions of every prescribed order $\rho$ with $0\le\rho\le\infty$. Hayman's 1960 theorem gives rays, so $\ell(r)=r$, whenever $\log M(r,f)=O((\log r)^2)$; Theorem 1 shows that no unbounded multiplicative relaxation of that hypothesis restores a linear-length path. The proofs build spiral barriers of small values by polynomial approximation and pass to an infinite product with controlled zero counting; the finite-order assertions and the barrier step are recorded, the theorems with proof sketches, on the result pages theorem_1, theorem_2 and spiral_barriers of the source card goldberg_1979_asymptotic_curves_entire_functions_finite_order. Theorem 4 (pp. 524--529), on paths to a finite asymptotic value, concerns the variant that Hayman's Problem 2.41 adds and not this problem's question.

Scope. The problem also asks for a path of slowest growth and an estimate of $\ell(r)$ in terms of $M(r)$. The paper supplies no optimal bound for every growth class, and none is claimed here: the claim settles the problem's definite question, whether $\ell(r)\ll r$ can always be achieved, and the sharpness of Hayman's growth threshold. Positive upper bounds in the literature, such as the finite-order bound the Hayman--Lingham survey reports under a first-intersection length, are recorded on the problem page and not asserted here.

Source. A. A. Gol'dberg and A. E. Eremenko, Asymptotic curves of entire functions of finite order (Russian), Mat. Sb. (N.S.) 109(151) (1979), no. 4, 555--581, 647; English translation, On asymptotic curves of entire functions of finite order, Math. USSR-Sb. 37 (1980), no. 4, 509--533, DOI 10.1070/SM1980v037n04ABEH001989; received 20 September 1977. The translation is the version cited here. The page is named by the original's year, which the record gives alone.

Acceptance. Refereed: the paper appeared in Matematicheskii Sbornik, with its translation in Mathematics of the USSR-Sbornik. Reviewed: the site's curator, T. F. Bloom, labels the problem solved and records that Gol'dberg and Eremenko disproved it with the functions of Theorems 1 and 2; Hayman and Lingham's 2018 survey of Hayman's problems (Update 2.41) records the problem as completely solved by this paper. Toppila's 1980 note, which the survey cites as an independent proof and which acknowledges this paper's priority, has its own claim page. Nothing here is independently reviewed by this project.

Formalization. The file Erdos1115.lean in Boris Alexeev's repository, added on 17 August 2026 and linked above at the commit of 23 August 2026 that carries its header, declares itself a Lean formalization of a solution to the problem, names A. A. Gol'dberg and Alexandre Eremenko as the informal authors and Codex and GPT-5.6 Sol as the formal authors, and proves erdos_1115: for every $\phi\to\infty$ there is a nonconstant entire function $f$ of finite order, in the file's elementary growth sense, with $\log M(r,f)\le C\phi(r)(\log r)^2$ for large $r$, such that no asymptotic path to infinity, parametrized with speed at most one, has length $O(r)$ inside the disc of radius $r$; the file also keeps the construction's escaping barriers in the conclusion. Its header says that the reconstruction of the paper and the correspondence with the file are in a TeX file of the repository. This corpus has not built or audited the development, so no formalized evidence is listed; the site shows no formalized statement for the problem.

Depends on. Nothing on the wiki. The proofs use Runge-type polynomial approximation and standard value-distribution estimates, cited on the result pages.