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Herzog and Piranian answer the first question of Problem 1117 affirmatively: there is an entire function $f$, not a monomial, for which the number $\nu(r)$ of points on $|z|=r$ where $|f|$ attains its maximum modulus satisfies $\limsup_{r\to\infty}\nu(r)=\infty$. The statement follows the site's commentary and Hayman and Lingham's survey of Hayman's problems, which records it as Update 2.16.

Covers. The first question, whether $\limsup\nu(r)=\infty$ is possible. The second question, whether $\liminf\nu(r)=\infty$ is possible, is outside this claim; a pending negative answer to it is recorded on Gu's page.

Standing. The site's curator, Thomas F. Bloom, credits Herzog and Piranian in the problem's commentary with the affirmative answer to the first question, and Hayman and Lingham's survey of Hayman's problems ([[../library/polynomials/hayman_lingham_2018_research_problems_function_theory/_index|library card]], Update 2.16) records the same. The site labels the problem OPEN, so the commentary is not an acceptance of the problem or of a part. The paper is F. Herzog and G. Piranian, The counting function for points of maximum modulus, in Entire Functions and Related Parts of Analysis, Proc. Sympos. Pure Math. 11, Amer. Math. Soc. (1968), 240--243. This is a symposium volume, so refereed is not listed, and no formalization is recorded, so the claim stays claimed. The page is dated by the publication year alone, since the record gives no finer date.

Depends on. Nothing beyond the cited paper.