Problem 513

../

claims/: The 4 claim pages of Problem 513, one per claimant's result; the problem's standing derives from them.

---

Statement. Let $f=\sum_{n=0}^\infty a_nz^n$ be a transcendental entire function. What is the greatest possible value of

$$\liminf_{r\to \infty} \frac{\max_n\lvert a_nr^n\rvert}{\max_{\lvert z\rvert=r}\lvert f(z)\rvert}?$$

Status. Open, the site's label (page last edited 2 April 2026). The value asked for, the supremum $B$, is known to lie in an interval whose ends are recorded below. One accepted partial claim, [[problems/analysis/E0513/claims/1964_12_01_clunie_hayman|Clunie and Hayman 1964]], proves $4/7<B\le2/\pi-c$ on its refereed publication; three pending partial claims certify larger lower bounds by interval arithmetic: [[problems/analysis/E0513/claims/2026_02_12_he_tang|He and Tang's certified lower bound]], an arXiv paper of 12 February 2026 announced in the site's discussion thread and credited by the site's commentary, which raised the bound from $4/7$ to $0.58507$; [[problems/analysis/E0513/claims/2026_02_27_sothanaphan|Sothanaphan's certified parameter improvement]], a note of 27 February 2026 produced with GPT-5.2 Thinking, announced in the discussion thread and credited by the commentary, which holds the best lower bound; and [[problems/analysis/E0513/claims/2026_08_02_lystad|Lystad's certified lower bound]], filed on the site's proof-claims tab on 2 August 2026 with declared AI assistance, which certifies a bound slightly below it and re-verifies it. The site has adopted none as a solution, and the derived standing is open.

Source. erdosproblems.com/513, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #513, https://www.erdosproblems.com/513.

References.

• [ClHa64] Clunie, J. and Hayman, W. K., The maximum term of a power series. J. Analyse Math. (1964), 143-186.
• [GrSh63] Gray, Alfred and Shah, S. M., A note on entire functions and a conjecture of Erdős. Bull. Amer. Math. Soc. (1963), 573-577.
• [HeTa26] Y. He and Q. Tang, Generalizing the Clunie-Hayman construction in an Erdős maximum-term problem. arXiv:2602.12217 (2026).

Formalization. Statement in formal-conjectures.

Current assessment

The question (site formulation of 2026-09-04; page last edited 2 April 2026). The statement above: with $\mu(r,f)=\max_n\lvert a_nr^n\rvert$ the maximum term and $M(r,f)$ the maximum modulus of a transcendental entire $f$ on $\lvert z\rvert=r$, the question asks for the largest possible value of $\liminf_{r\to\infty}\mu(r,f)/M(r,f)$; write $B$ for the supremum of these values over all such $f$. The label is OPEN; the site cross-references Problem 227. The discussion thread holds six comments: two of December 2025 agreeing that the statement needs the word transcendental, which Clunie and Hayman assumed; Tang's announcement of 13 February 2026 of the paper [HeTa26]; Tao's of the same day of a page for the constant $B$ in his repository of optimization constants; Sothanaphan's of 1 March 2026 announcing the computation recorded below; and the curator's of 2 April 2026 on citing the upper bound $2/\pi-c$.

Known results. As the site's commentary records them: $1/2\le B\le1$ is elementary; Kövári observed, unpublished, that $B>1/2$; an argument of Clunie reported by Gray and Shah [GrSh63] gives $B\le2/\pi\approx0.63662$; Clunie and Hayman [ClHa64] proved $4/7<B\le2/\pi-c$ for an absolute $c>0$, with $4/7\approx0.57143$, the accepted partial claim [[problems/analysis/E0513/claims/1964_12_01_clunie_hayman|Clunie and Hayman 1964]], the best refereed bounds on $B$; the Gray and Shah note has no claim page, because its bound $B\le2/\pi$ is Clunie's argument reported by the two authors and is contained in the bound Clunie and Hayman published the following year. He and Tang [HeTa26] raised the lower bound to $B>0.58507$ (their Theorem 1.2; the commentary prints $0.5850724$) by generalizing the Clunie-Hayman construction to a two-parameter family $f_{K,\epsilon}$ with an exact formula for the limit inferior along the radii $K^m$ and a ball-arithmetic certificate of a unit-circle maximum. The commentary further credits a slight improvement to $B>0.5850788$ to a computation by GPT, as the commentary names the system, prompted by Sothanaphan. That computation is Sothanaphan's note of 27 February 2026, which names GPT-5.2 Thinking, announced in the discussion thread on 1 March 2026 and recorded on its claim page, [[problems/analysis/E0513/claims/2026_02_27_sothanaphan|Sothanaphan's certified parameter improvement]]: an interval-arithmetic certificate for a new parameter choice in the He-Tang family giving $B\ge0.585078819653$, of which the commentary's figure is a truncation. The best recorded bounds are thus $0.585078819653\le B\le2/\pi-c$, and the value of $B$ is open.

Pending claims. He and Tang's paper, recorded above, is a dated arXiv manuscript announced in the discussion thread and not on the proof-claims tab; it is unrefereed, and the commentary's credit on a problem the site labels OPEN is not an acceptance, so it stays claimed on its page, [[problems/analysis/E0513/claims/2026_02_12_he_tang|He and Tang's certified lower bound]]. Sothanaphan's note, recorded above, is a dated manuscript posted in the discussion thread and not on the proof-claims tab; it is unrefereed on the same rule and stays claimed. The tab's one entry, a partial proof claim filed 2 August 2026, is [[problems/analysis/E0513/claims/2026_08_02_lystad|Lystad's certified lower bound]]: $B\ge0.5850788196$ from an explicit member of the He-Tang family, certified by interval arithmetic, together with an independent re-verification of Sothanaphan's bound. It does not move the frontier and does not touch the upper bound; it is unreviewed and stays claimed.

Search scope. The site's problem page and proof-claims tab (2026-10-06), its discussion thread (2026-10-07), Sothanaphan's note, the description of Lystad's Zenodo record, and the library card of [HeTa26]. No refereed work beyond the references was found.

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.

• he_2026_generalizing_clunie_hayman_construction_erdos_maximum
• he_2026_generalizing_clunie_hayman_construction_erdos_maximum / proposition_3_5
• he_2026_generalizing_clunie_hayman_construction_erdos_maximum / theorem_1_2
• he_2026_generalizing_clunie_hayman_construction_erdos_maximum / theorem_2_8