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Claim. The answer to Problem 516 is yes. W. H. J. Fuchs, Proof of a conjecture of G. Pólya concerning gap series, Illinois J. Math. 7 (1963), no. 4, 661--667, doi:10.1215/ijm/1255645102, proves that if $f(z)=\sum_{k\ge1}a_kz^{n_k}$ is an entire function of finite order with $n_k/k\to\infty$, then for every $\epsilon>0$

$$\log m(r)>(1-\epsilon)\log M(r)$$

holds for all $r$ outside a set of logarithmic density zero, where $M(r)$ and $m(r)$ are the maximum and minimum modulus of $f$ on $\lvert z\rvert=r$. Since the exceptional set has logarithmic density zero, radii with $\log m(r)/\log M(r)>1-\epsilon$ exist beyond every bound, and the trivial inequality $m(r)\le M(r)$ gives the reverse bound, so

$$\limsup_{r\to\infty}\frac{\log m(r)}{\log M(r)}=1,$$

which is the question's statement, with the larger assertion about density as a bonus. The problem is Pólya's; the statement above follows the site's account of the paper; the proof is not reconstructed in this repository. Erdős's 1961 formulation, with $m(r)$ the maximum term instead of the minimum modulus, is a different and much easier statement; the problem page records that reading.

Acceptance. Refereed: the paper appeared in the Illinois Journal of Mathematics, a refereed journal; the publisher's record (Crossref) dates the issue 1 December 1963, which is this page's date. Reviewed: the site's curator, Thomas Bloom, labels the problem PROVED (LEAN) and credits this paper with the affirmative solution on erdosproblems.com/516 (page last edited 28 December 2025), with the community database in agreement, which records the problem as proved. A Lean development in Boris Alexeev's lean-proofs repository, linked above at the revision of 15 September 2026 that the formal-conjectures statement file points to, declares itself a formalization of a solution to the problem, names Fuchs as its informal author, the Formal Conjectures authors as the statement authors, and Codex and GPT-5.6 Sol as its formal authors, and builds on Lean 4.33.0 with Mathlib; the community database records a Lean proof. That development has not been built or audited in this repository, so it is a link and not formalized evidence. No independent review is recorded here and none is claimed.

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