Problem 1117

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claims/: The 2 claim pages of Problem 1117, one per claimant's result; the problem's standing derives from them.

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Statement. Let $f(z)$ be an entire function which is not a monomial. Let $\nu(r)$ count the number of $z$ with $\lvert z\rvert=r$ such that $\lvert f(z)\rvert=\max_{\lvert z\rvert=r}\lvert f(z)\rvert$. (This is a finite quantity if $f$ is not a monomial.)

Is it possible for

$$\limsup \nu(r)=\infty?$$

Is it possible for

$$\liminf \nu(r)=\infty?$$

Status. The site labels the problem OPEN (page last edited 29 December 2025). The site's commentary records that the first question has the answer yes, by Herzog and Piranian (1968), a pending partial claim on their page, and that the second question is open, with an approximate affirmative analogue by Glücksam and Pardo-Simón [GlPa24]. The site's proof-claims tab carries a claim credited to Qiyuan Gu, submitted 2026-09-05 and listed there as a full claim; the claim's notes say that GPT-6 Astra generated the proofs and that GPT-5.6 Sol and Claude Opus 5 were used for editorial review. It asserts $\nu(r)\le2k$ outside a countable set of radii, where $k$ is the gap between the exponents of the first two nonzero terms of $f$, so that $\liminf\nu(r)<\infty$ for every non-monomial entire $f$, a negative answer to the second question. It is pending on its page as a partial claim settling the second question (proof-claims thread accessed 2026-10-06). The derived standing, claimed with the value answered, departs from OPEN because a pending claim answers the second question, which the site's commentary records as open, and with the pending affirmative answer to the first every part is covered by a pending claim.

Source. erdosproblems.com/1117, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1117, https://www.erdosproblems.com/1117.

References.

• [GlPa24] Glücksam, Adi and Pardo-Simón, Leticia, An approximate solution to Erdős' maximum modulus points problem. J. Math. Anal. Appl. 531 (2024), no. 1, Paper No. 127768, 20.
• [Ha74] Hayman, W. K., Research problems in function theory: new problems. (1974), 155-180.
• [HePi68] Herzog, F. and Piranian, G., The counting function for points of maximum modulus. (1968), 240-243.

Formalization. None recorded.

Current assessment

First question answered yes; second question claimed no, pending. The problem asks whether a non-monomial entire function can have $\limsup\nu(r)=\infty$, and whether it can have $\liminf\nu(r)=\infty$. Herzog and Piranian [HePi68] construct an entire function with $\nu(n)=n$ for every natural $n$, which answers the first question yes; the site's commentary and Hayman and Lingham's survey of Hayman's problems credit them, but the paper is in a symposium volume, so it is a pending partial claim on their page. Their construction gives no control of $\nu(r)$ between integers. For the second question, Glücksam and Pardo-Simón [GlPa24] construct an entire function for which, for every $\varepsilon>0$, the number of components of $\{|z|=r\}$ meeting the $\varepsilon$-approximate maximum modulus set tends to infinity; this concerns approximate rather than exact maximum modulus points, so it settles no part and is not a claim. Gu's manuscript asserts $\nu(r)\le2k$ outside a countable set of radii, a negative answer to the second question; it is unrefereed and unreviewed, and is a pending partial claim on its page. With both parts covered by pending claims, the problem's derived standing is claimed, answered.

Search scope. The site's page and commentary (last edited 29 December 2025), its proof-claims tab (accessed 2026-10-06), the Zenodo record of Gu's manuscript, and the library cards of [GlPa24] and of Hayman and Lingham's survey; [HePi68] and [Ha74] are cited from the site and that survey.

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.

• glucksam_2024_approximate_solution_erdos_maximum_modulus_points
• glucksam_2024_approximate_solution_erdos_maximum_modulus_points / question_1_1
• glucksam_2024_approximate_solution_erdos_maximum_modulus_points / theorem_1_2
• hayman_lingham_2018_research_problems_function_theory
• hayman_lingham_2018_research_problems_function_theory / problem_2_16