Problem 516

Let $f(z)=\sum_{k\geq 1}a_k z^{n_k}$ be an entire function of finite order such that $\lim n_k/k=\infty$ . Let $M(r)=\max_{\lvert z\rvert=r}\lvert f(z)\rvert$ and $m(r)=\min_{\lvert z\rvert=r}\lvert f(z)\rvert$ . Is it true that

\limsup\frac{\log m(r)}{\log M(r)}=1?

Source: erdosproblems.com/516

Analysis

Wiki page

Status

SolvedProved

An accepted solution exists. The statement is true.

PROVED (LEAN). The site labels the problem PROVED (LEAN) (page last edited 28 December 2025) and credits Fuchs [Fu63] with the affirmative solution; the Lean qualifier refers to a proof in Boris Alexeev's lean-proofs repository that has not been built here (see Formalization). The accepted claim page Fuchs 1963 records the result, on the refereed venue and the site's acceptance, and carries the formalization link; the frontmatter standing derives from it. Two earlier refereed results settle subclasses and are accepted partial claims: Erdős and Macintyre 1954 and Kővári 1965.

Er61: Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254.
ErMa54: Erdős, P. and Macintyre, A. J., Integral functions with gap power series. Proc. Edinburgh Math. Soc. (2) (1954), 62-70.
Fu63: Fuchs, W. H. J., Proof of a conjecture of G. Pólya concerning gap series. Illinois J. Math. (1963), 661-667.
Ko65: Kövari, Thomas, A gap-theorem for entire functions of infinite order. Michigan Math. J. (1965), 133-140.
Ma52: Macintyre, A. J., Asymptotic paths of integral functions with gap power series. Proc. London Math. Soc. (3) (1952), 286-296.
Po29: Pólya, G., Untersuchungen über Lücken und Singularitäten von Potenzreihen. Math. Z. (1929), 549-640.
Wi14: Wiman, A., Über den Zusammenhang Zwischen dem Maximalbetrage Einer Analytischen Funktion und dem Grössten Gliede der Zugehörigen Taylor'schen Reihe. Acta Math. (1914), 305-326.

Problem 516

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Proof claims

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