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Claim. The answer to Problem 515 is yes. John Lewis, John Rossi and Allen Weitsman, On the growth of subharmonic functions along paths, Ark. Mat. 22 (1984), no. 1--2, 109--119, doi:10.1007/BF02384375, prove that for every entire function $f$ that is not a polynomial there is a locally rectifiable path $C$ tending to infinity on which

$$\int_C \lvert f(z)\rvert^{-\lambda}\,\lvert\mathrm{d}z\rvert<\infty \quad\text{for every }\lambda>0,$$

one path serving every exponent at once, the integral being taken with respect to arc length, as the paper writes it. This is the case $u=\log\lvert f\rvert$ of the paper's Theorem 1: for a function $u$ subharmonic in the plane with $M(r,u)/\log r\to\infty$ as $r\to\infty$, where $M(r,u)=\max_{\lvert z\rvert=r}u(z)$, there is a path $\Gamma$ tending to infinity with $\int_\Gamma e^{-\lambda u}\,\lvert\mathrm{d}z\rvert<\infty$ for each $\lambda>0$ and $u(z)/\log\lvert z\rvert\to\infty$ as $z\to\infty$ along $\Gamma$. For $u=\log\lvert f\rvert$ the growth hypothesis says that $f$ is not a polynomial, since $\log M(r,f)/\log r$ stays bounded exactly when $f$ is one. The paper says that the point of the theorem is that the path does not depend on $\lambda$, that it answers a question of Hayman, and that Zhang had proved the case $u=\log\lvert f\rvert$ with $f$ of finite order. The statement above follows its Theorem 1 and introduction. Earlier partial results are recorded on the problem page: Huber found, for each fixed $\lambda$, a path on which that one integral is finite, and Zhang settled the case of finite order, the accepted partial claim Zhang 1977. The path-growth clause of Theorem 1 also answers the first question of Problem 514, recorded on [[problems/analysis/E0514/claims/1984_12_01_lewis_rossi_weitsman|its claim page there]].

Acceptance. Refereed: the paper appeared in Arkiv för Matematik, a refereed journal; the publisher's record (Crossref) dates the issue December 1984, and this page's date is the first day of that month. Reviewed: the site's curator, Thomas Bloom, labels the problem PROVED and credits this paper with the general case on erdosproblems.com/515 (page last edited 19 October 2025), with the community database in agreement, which records the problem as proved. No independent review is recorded in this repository and none is claimed.

Depends on. Nothing in this wiki: the argument is the paper's own.