Problem 515

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claims/: The 2 claim pages of Problem 515, one per claimant's result; the problem's standing derives from them.

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Statement. Let $f(z)$ be an entire function, not a polynomial. Does there exist a locally rectifiable path $C$ tending to infinity such that, for every $\lambda>0$, the integral

$$\int_C \lvert f(z)\rvert^{-\lambda} \mathrm{d}z$$

is finite?

Status. Proved. The site labels the problem PROVED (page last edited 19 October 2025) and credits Lewis, Rossi and Weitsman [LRW84] with the general case, after Zhang [Zh77] had settled the entire functions of finite order and Huber [Hu57] had found, for each fixed $\lambda$, a path on which that one integral is finite. The accepted claim page Lewis, Rossi and Weitsman 1984 records the result, on the refereed venue and the site's acceptance; the frontmatter standing derives from it. Zhang's finite-order case is the accepted partial claim Zhang 1977, on its journal publication.

Source. erdosproblems.com/515, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #515, https://www.erdosproblems.com/515.

References.

• [Hu57] Huber, Alfred, On subharmonic functions and differential geometry in the large. Comment. Math. Helv. (1957), 13-72.
• [LRW84] Lewis, John and Rossi, John and Weitsman, Allen, On the growth of subharmonic functions along paths. Ark. Mat. (1984), 109-119.
• [Zh77] Zhang, Guang Hou, Asymptotic values of entire and meromorphic functions. Kexue Tongbao (1977), 480, 486.

Formalization. None recorded.

Current assessment

The question, in the site's formulation of 2026-09-04, asks for one locally rectifiable path to infinity on which $\lvert f\rvert^{-\lambda}$ is integrable for every $\lambda>0$ at once, for every entire $f$ that is not a polynomial. The answer is yes: the general case is the 1984 theorem of Lewis, Rossi and Weitsman, accepted here on its refereed publication and the site's credit, and recorded on its claim page. The statement of [LRW84] follows its Theorem 1 and introduction, and the other statements follow the site's account and the publishers' records; the proof is not reconstructed in this repository and no independent review is recorded. Huber's paths, one for each exponent, settle no instance of the question and have no claim page. The status search covered the site, its forum thread and the community database on 2026-10-07; the thread holds no proof claim, and no other claim of the result was found.

Known Results

• Huber [Hu57] proved the one-exponent version: for each fixed $\lambda>0$ there is a path $C_\lambda$ tending to infinity on which $\int_{C_\lambda}\lvert f\rvert^{-\lambda}$ is finite. The path may depend on $\lambda$, so this does not answer the question.
• Zhang [Zh77] proved the question's statement for entire functions of finite order; the paper Lewis, Rossi and Weitsman cite for it is Zhang Guanghou, Asymptotic values of entire and meromorphic functions, Sci. Sinica 20 (1977), 720-739, of the same title and year as the site's Kexue Tongbao item. This is the accepted partial claim Zhang 1977.
• Lewis, Rossi and Weitsman [LRW84] proved it for every nonpolynomial entire function, as the case $u=\log\lvert f\rvert$ of their Theorem 1, which gives a path independent of $\lambda$ for every function $u$ subharmonic in the plane with $\max_{\lvert z\rvert=r}u(z)/\log r\to\infty$, a growth hypothesis that for $u=\log\lvert f\rvert$ says that $f$ is not a polynomial. This is the accepted claim Lewis, Rossi and Weitsman 1984.