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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. Nat Sothanaphan, A certified computation for an improved He-Tang parameter choice in Erdős' maximum-term problem, a note dated 27 February 2026, proves in its Theorem 1 that for $K=3.568182317714$ and $\varepsilon=e^{i\alpha}$ with $\alpha=3.961543335688$ the function $f_{K,\varepsilon}$ of He and Tang's two-parameter family satisfies $\max_{\lvert z\rvert=1}\lvert k_{K,\varepsilon}(z)\rvert\le1.709171425130$, where $k_{K,\varepsilon}$ is the associated Laurent series, and hence that

$$\liminf_{r\to\infty}\frac{\mu(r,f_{K,\varepsilon})}{M(r,f_{K,\varepsilon})}\ge0.585078819653, \qquad\text{so}\qquad B\ge0.585078819653.$$

The reduction of the limit inferior to the reciprocal of that unit-circle maximum is Theorem 2.8 of He and Tang's paper, whose card is he_2026_generalizing_clunie_hayman_construction_erdos_maximum; the note's contribution is the parameter choice and its certificate. On the unit circle the series is a cosine series, which the note truncates with a geometric bound on the tail; the maximum of the truncation is certified in interval arithmetic by a dyadic cover of the critical points under a global Lipschitz bound on the derivative, with $K$ and $\alpha$ read as exact rationals from their decimal expansions. The note reports that the same code run on He and Tang's own parameters certifies the bound $0.585077117559$, slightly above their published $0.58507$, and it records a certified computation for a free-phase relaxation of the cosine series that, it says, corresponds to no proven entire-function construction. The note's disclaimer says that the document was generated in a near-autonomous process by GPT-5.2 Thinking, and a code package accompanies it.

Submission note. Posted to the site's forum by Nat Sothanaphan on 1 March 2026:

I have run GPT in a near-autonomous process to try to improve the constant. Here is the writeup.

We did not really succeed. To summarize, the bound $B > 0.5850724$ in He-Tang is very slightly improved to $B > 0.5850788$ (both numbers rounded down to nearest). More ambitious improvements were attempted but they did not succeed.

I was debating with myself whether this should count as an improvement. But seeing the AlphaEvolve's precedents in [36] (also recorded here) and [1097] which are of similar scales, I believe it should count.

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

Covers. The lower bound $B\ge0.585078819653$ only. The value of $B$ and the upper bound $B\le2/\pi-c$ of Clunie and Hayman are not addressed.

Standing. Sothanaphan announced the note in the site's discussion thread on 1 March 2026, describing the computation as GPT run in a near-autonomous process. The site's commentary (page last edited 2 April 2026) credits the improvement of the lower bound to $0.5850788$, a truncation of the note's value, to GPT as prompted by Sothanaphan, and Tao's table of bounds for the constant records the value with the same credit. The problem's label is OPEN, so the commentary's credit is not an acceptance. The note is not refereed and not formalized, and the only check of it recorded anywhere is the re-verification claimed in [[problems/analysis/E0513/claims/2026_08_02_lystad|Lystad's certified lower bound]], itself unreviewed. The claim stays claimed.

Depends on. [[problems/analysis/E0513/claims/2026_02_12_he_tang|He and Tang's certified lower bound]], whose paper supplies the reduction of the limit inferior to the unit-circle maximum (their Theorem 2.8) and is unrefereed; the certificate is the note's own.