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Claim. Let $B$ be the supremum, over transcendental entire functions $f$, of $\liminf_{r\to\infty}\mu(r,f)/M(r,f)$, where $\mu(r,f)$ is the maximum term of the power series of $f$ at radius $r$ and $M(r,f)$ its maximum modulus there; $B$ is the value Problem 513 asks for. Yixin He and Quanyu Tang, Generalizing the Clunie-Hayman construction in an Erdős maximum-term problem, arXiv:2602.12217, prove in their Theorem 1.2 that there are parameters $K_0>1$ and $\lvert\varepsilon_0\rvert=1$ for which the transcendental entire function $f_{K_0,\varepsilon_0}$ of their two-parameter family $f_{K,\varepsilon}(z)=\sum_{n\ge0}\varepsilon^{n(n-1)/2}K^{-n(n+1)/2}z^n$ satisfies

$$\liminf_{r\to\infty}\frac{\mu(r,f_{K_0,\varepsilon_0})}{M(r,f_{K_0,\varepsilon_0})}>0.58507, \qquad\text{so}\qquad B>0.58507,$$

improving the bound $4/7$ of Clunie and Hayman. The family generalizes the Clunie-Hayman construction by replacing its sign pattern $(-1)^{n(n-1)/2}$ with an arbitrary unimodular phase. The associated Laurent series $k_{K,\varepsilon}$ obeys the scaling identity $k_{K,\varepsilon}(Kz)=z\,k_{K,\varepsilon}(\varepsilon z)$, which gives the maximum modulus of $k_{K,\varepsilon}$ and the maximum term of $f_{K,\varepsilon}$ exactly along the radii $K^m$ and reduces the limit inferior to the reciprocal of $\max_{\lvert z\rvert=1}\lvert k_{K,\varepsilon}(z)\rvert$ (their Theorem 2.8). The parameters are $K_0=7137/2000$ and $\varepsilon_0=e^{i\alpha_0}$ with $\alpha_0=198074929/50000000$; on the unit circle the series is a cosine series, which the paper truncates after six terms with a geometric bound on the tail, and the truncation's values on a mesh of $2\times10^6$ points are certified in Arb ball arithmetic to have modulus at most $1.709176398$ (their Lemma 3.4); since the truncation is $4$-Lipschitz (their Lemma 3.3, from a bound on its derivative), its maximum is below $1.709183$, and adding the tail bound gives $\max_{\lvert z\rvert=1}\lvert k_{K_0,\varepsilon_0}(z)\rvert<1.70919$, with $K_0$ and $\alpha_0$ read as exact rationals (their Proposition 3.5 and Appendix A, which holds the code and logs). The paper's declaration of AI usage names ChatGPT, model GPT-5.2 Pro, as used for exploratory brainstorming and for drafting the first version of one certification script, with every argument and all code checked by the authors. The statement follows the paper, whose card is he_2026_generalizing_clunie_hayman_construction_erdos_maximum.

Submission note. Posted to the site's forum by Quanyu Tang on 13 February 2026:

Let

$$> B:=\sup_f \ \liminf_{r\to\infty}\frac{\max_n |a_n|\,r^n}{\max_{|z|=r}|f(z)|}. >$$

1. In addition to the upper bound $B<2/\pi$, [ClHa64] also proved the lower bound $B>4/7$.

2. This problem is also listed in [HaLi18] (see arxiv link, springer link), where it is recorded that the best known bounds are

$$> 0.57143\approx \frac{4}{7} < B < \frac{2}{\pi}\approx 0.63662. >$$

3. My friend He and I have just posted a paper (arXiv:2602.12217) in which we generalize the function construction from [ClHa64] and obtain a stronger explicit lower bound

$$> B>0.58507. >$$

This work made exploratory use of ChatGPT-5.2 Pro, with all mathematical arguments subsequently checked by the human authors.

Reference. [HaLi18] W. K. Hayman, E. F. Lingham, Research Problems in Function Theory: Fiftieth Anniversary Edition, Problem Books in Mathematics, Springer, Cham, 2019.

(The site has been updated to address this comment.)

Covers. The lower bound $B>0.58507$ only. The value of $B$ and the upper bound $B\le2/\pi-c$ of Clunie and Hayman are not addressed.

Standing. Tang announced the paper in the site's discussion thread on 13 February 2026. The site's commentary (page last edited 2 April 2026) credits the lower bound $0.5850724$ to the paper, and Tao's table of bounds for the constant records it with the same credit. The problem's label is OPEN, so the commentary's credit is not an acceptance. The paper is an arXiv preprint, not refereed and not formalized. Two checks of its certificate are recorded, both unreviewed: [[problems/analysis/E0513/claims/2026_02_27_sothanaphan|Sothanaphan's note]] reports that its own certification code run on He and Tang's parameters certifies the bound $0.585077117559$, and Lystad's record re-certifies the same parameters to the same bound as a regression preset of its own certificate. The claim stays claimed.

Depends on. Nothing among the wiki's pages; the argument is the paper's own.