Problem 514

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claims/: The 3 claim pages of Problem 514, one per claimant's result; the problem's standing derives from them.

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Statement. Let $f(z)$ be an entire transcendental function. Does there exist a path $L$ so that, for every $n$,

$$\lvert f(z)/z^n\rvert \to \infty$$

as $z\to \infty$ along $L$?

Can the length of this path be estimated in terms of $M(r)=\max_{\lvert z\rvert=r}\lvert f(z)\rvert$? Does there exist a path along which $\lvert f(z)\rvert$ tends to $\infty$ faster than a fixed function of $M(r)$ (such that $M(r)^\epsilon$)?

Formulation. The statement asks three questions, the parts path, length and growth of this page. Erdős [Er61] (IV.6, p. 249) states the third as growth faster than a fixed function of $M(r)$, giving $M(r)^\epsilon$ as an example; the site's display prints "such that" where such as is meant. The page reads the comparison function as fixed before the entire function is chosen, which is how both claimants read it: a function chosen after $f$ makes the question trivially yes, since $\lvert f\rvert\to\infty$ along the path of the first question and any slower function of $M(r)$ then works. The standing targets this reading, with the power $M(r)^\epsilon$ as its special case.

Status. The site labels the problem OPEN (page last edited 18 January 2026) and credits Boas, unpublished, with the first question. The three parts are answered by three partial claims: the first question yes, by the accepted claim [[problems/analysis/E0514/claims/1984_12_01_lewis_rossi_weitsman|Lewis, Rossi and Weitsman 1984]], on its refereed publication; the first and second yes, by the pending claim Chojecki 2026, a note of 20 April 2026 produced with GPT-5.4 Pro; and the third no, by the pending claim [[problems/analysis/E0514/claims/2026_04_28_oriike|Oriike 2026]], a note of 28 April 2026 produced with GPT-5.5 Pro, with a Lean file. Accepted and pending claims together settle every part, so the derived standing is claimed, and its value is answered because the parts are answered in opposite directions. The site has adopted none of the claims.

Source. erdosproblems.com/514, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #514, https://www.erdosproblems.com/514.

References.

• [Ch26] Chojecki, P., A note on an Erdős path problem for transcendental entire functions. Note (2026), https://www.ulam.ai/research/erdos514.pdf.
• [Er61] Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254.
• [Er80] Eremenko, A. È., Growth of entire and subharmonic functions on asymptotic curves. Sib. Math. J. 21 (1980), no. 5, 673-683.
• [GoEr80] Gol'dberg, A. A. and Eremenko, A. È., On asymptotic curves of entire functions of finite order. Math. USSR-Sb. 37 (1980), 509-533.
• [Ha60] Hayman, W. K., On the growth of integral functions on asymptotic paths. J. Indian Math. Soc. (N.S.) 24 (1960), 251-264.
• [La22] Langley, J. K., Complex flows, escape to infinity and a question of Rubel. Ann. Fenn. Math. 47 (2022), 885-894.
• [LRW84] Lewis, John and Rossi, John and Weitsman, Allen, On the growth of subharmonic functions along paths. Ark. Mat. (1984), 109-119.
• [Or26] Oriike, Yuta, A negative answer to the universal-function version of Erdős's third question in Problem 514. Note (2026), revised May 2026.
• [Ta76] Talpur, M. N. M., On the growth of subharmonic functions on asymptotic paths. Proc. London Math. Soc. (3) 32 (1976), no. 2, 193-198.
• [Wu85] Wu, Jang-Mei, Length of paths for subharmonic functions. J. London Math. Soc. (2) 32 (1985), 497-505.

Formalization. No statement file exists in formal-conjectures, and the community database records no formalized statement. A Lean file accompanying Oriike's note declares itself a formalization of that note's theorem; it is linked from the claim page and has not been built or audited in this repository.

Current assessment

The question (site formulation of 2026-09-04; page last edited 18 January 2026). For a transcendental entire function $f$ with maximum modulus $M(r)$: does a path to infinity exist on which $\lvert f(z)/z^n\rvert\to\infty$ for every $n$ (path); can the length of such a path be bounded in terms of $M(r)$ (length); and does a path exist on which $\lvert f\rvert$ grows faster than a fixed function of $M(r)$, such as $M(r)^\epsilon$ (growth)? The Formulation above fixes the reading of the third question. The label is OPEN; the site credits Boas, unpublished, with the first question, as [Er61] does, where Erdős says he conjectured it and Boas proved it without publishing; the statement was given the word transcendental after a thread comment of 17 January 2026.

Answers. The first question is answered yes in refereed print by Theorem 1 of Lewis, Rossi and Weitsman [LRW84], applied to $u=\log\lvert f\rvert$: there is a path on which $\log\lvert f(z)\rvert/\log\lvert z\rvert\to\infty$, which is the accepted partial claim [[problems/analysis/E0514/claims/1984_12_01_lewis_rossi_weitsman|Lewis, Rossi and Weitsman 1984]]; the same theorem settles Problem 515. Lewis, Rossi and Weitsman present Theorem 1 as a generalization of Huber's Theorem A, one path for each exponent, and of Talpur's Theorem B [Ta76], which already gives, for every $u$ subharmonic in the plane with $M(r,u)/\log r\to\infty$, a path to infinity on which $u(z)/\log\lvert z\rvert\to\infty$; so the first question was already answered in refereed print by Talpur's theorem with $u=\log\lvert f\rvert$, and the 1984 paper's addition is the integral condition on the same path. Talpur's paper has no claim page: the site does not credit it, and its theorem is the path clause of the accepted claim. Boas's proof has no manuscript and no page. The second question is answered yes by Theorem 1 of Chojecki's note [Ch26]: the path given by that theorem in Wu's restatement [Wu85] (Theorem B) has initial segments of length $O(M(R,f)^\epsilon)$ for every $\epsilon>0$, deduced from the convergence of $\int e^{-\epsilon u}\lvert\mathrm{d}z\rvert$ along the path; this is the pending claim Chojecki 2026, which also answers the first question. The third question is answered no by Theorem 1 of Oriike's note [Or26]: for every nondecreasing $\Phi\to\infty$ there is a transcendental entire $f$ with $\liminf\lvert f\rvert/\Phi(M_f)=0$ along every path to infinity; the revised note derives this from Hayman's theorem [Ha60] (Theorem 2) on growth along asymptotic paths through a selection lemma, keeping its direct construction as a second proof; this is the pending claim Oriike 2026. Chojecki's Theorem 2 refutes the power example $M(r)^\epsilon$ independently, from Langley's theorem [La22], but by its own Remark 7 not the fixed-function reading. Hayman's paper has no claim page: the site does not credit it, its theorem refutes the power example directly only as the two notes read it, and the deduction of the fixed-function answer is Oriike's.

Other inputs. A thread post of 7 August 2026 by Eremenko names two papers, Gol'dberg and Eremenko [GoEr80] and Eremenko [Er80], as essentially settling the length and growth questions for entire functions of finite order; the post states no theorem, the site does not credit the papers, and they are recorded here without a claim page. The joint paper of Chojecki and Oriike announced in the thread on 5 June 2026 is not recorded as a preprint. Sothanaphan's thread checks of the two notes (20 April, 29 April and 10 May 2026), and his streamlined derivation of Chojecki's result produced with GPT-5.5 Thinking, are recorded on the two claim pages as thread posts, not as review evidence. The proof-claims tab is empty. No proof is reconstructed in this repository and no independent review is recorded.

Search scope. The site's problem page, its discussion thread and proof-claims tab, and the community database (2026-10-07); the two notes and Oriike's Lean file at the revisions the claim pages link; arXiv for the announced joint paper (2026-10-07, none found).

Known results

• Lewis, Rossi and Weitsman [LRW84] prove, for every subharmonic $u$ in the plane with $\max_{\lvert z\rvert=r}u/\log r\to\infty$, a path to infinity on which $u(z)/\log\lvert z\rvert\to\infty$ and $\int e^{-\lambda u}\lvert\mathrm{d}z\rvert<\infty$ for every $\lambda>0$. With $u=\log\lvert f\rvert$ this answers the first question yes: the accepted partial claim [[problems/analysis/E0514/claims/1984_12_01_lewis_rossi_weitsman|Lewis, Rossi and Weitsman 1984]].
• Wu [Wu85] restates that theorem as Theorem B and sharpens the length estimates for paths of subharmonic functions of finite lower order (recorded on the library card).
• Chojecki [Ch26] answers the first two questions yes, with the length bound $O(M(R,f)^\epsilon)$ for every $\epsilon>0$, and refutes the power example of the third: the pending partial claim Chojecki 2026.
• Oriike [Or26] answers the third question no for every fixed comparison function, from Hayman [Ha60] and by a direct construction: the pending partial claim Oriike 2026.
• Gol'dberg and Eremenko [GoEr80] and Eremenko [Er80] study the length of asymptotic curves and the growth along them for entire functions of finite order; named in the thread, not assessed here.

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.

• chojecki_2026_note_erdos_path_problem_transcendental_entire
• chojecki_2026_note_erdos_path_problem_transcendental_entire / theorem_1
• chojecki_2026_note_erdos_path_problem_transcendental_entire / theorem_2
• oriike_2026_negative_answer_universal_function_version_erdos
• oriike_2026_negative_answer_universal_function_version_erdos / corollary_1
• oriike_2026_negative_answer_universal_function_version_erdos / theorem_1
• wu_1985_length_paths_subharmonic_functions
• wu_1985_length_paths_subharmonic_functions / theorem_2
• wu_1985_length_paths_subharmonic_functions / theorem_b