He 2026 generalizing clunie hayman construction erdos maximum

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proposition_3_5: He and Tang's certified bound that |k_{K_0,eps_0}| stays below 1.70919 on the unit circle for their chosen parameters, from a six-term truncation checked in ball arithmetic on a mesh of 2,000,000 points.

theorem_1_2: He and Tang's theorem that some function f_{K_0,eps_0} of their two-parameter Clunie-Hayman family has beta > 0.58507, so that Erdős's maximum-term constant B exceeds 0.58507.

theorem_2_8: He and Tang's exact formula: for every K > 1 and unimodular eps, the liminf of maximum term over maximum modulus for f_{K,eps} equals the reciprocal of the maximum of |k_{K,eps}| on the unit circle.

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Yixin He, Quanyu Tang, Generalizing the Clunie--Hayman construction in an Erdős maximum-term problem. arXiv:2602.12217 (2026). The arXiv record names arXiv's non-exclusive distribution license (arXiv:2602.12217), every other right reserved.

Erdos asked for the value of B = sup_f liminf_{r to infinity} mu(r,f)/M(r,f) over transcendental entire functions, where mu is the maximum term of the power series and M the maximum modulus; the paper recalls that Clunie and Hayman (1964) proved 4/7 < B < 2/pi. The authors generalize the Clunie-Hayman construction to a two-parameter family f_{K,eps}(z) = sum eps^{n(n-1)/2} K^{-n(n+1)/2} z^n with K > 1 and |eps| = 1, replacing the sign pattern (-1)^{n(n-1)/2} by the phase eps^{n(n-1)/2}. The associated Laurent series k_{K,eps} (the same sum over all integers n) obeys the scaling identity k(Kz) = z k(eps z) (Lemma 2.2), which gives the maximum modulus of k_{K,eps} exactly along the geometric radii r_m = K^m (Lemma 2.3); the maximum term of f_{K,eps} is also exact there (Lemma 2.4), the lim inf may be taken along these radii (Proposition 2.5), and f_{K,eps} differs from k_{K,eps} by O(1/|z|), yielding beta(f_{K,eps}) = 1 / max_{|z|=1} |k_{K,eps}(z)| (Theorem 2.8). With K_0 = 7137/2000 and eps_0 = e^{i alpha_0}, alpha_0 = 198074929/50000000, the paper bounds the unit-circle maximum A_0 by truncating the cosine series after six terms, certifying the maximum of the truncation on a mesh of 2,000,000 points with ball arithmetic, and adding a Lipschitz bound and a tail bound, so that A_0 < 1.70919 (Proposition 3.5); this gives Theorem 1.2: B > 0.58507, improving 4/7 ≈ 0.57143. Appendix A describes the mesh certification and records its output log; the code is in the second author's public repository. The paper's declaration of AI usage (Section 1.1, p. 2) says ChatGPT (GPT-5.2 Pro) was used for exploratory brainstorming and to draft the first version of the certification script, and that the authors checked all arguments and code.

Source: https://arxiv.org/abs/2602.12217.

Results. Theorem 1.2 (p. 2, proved on p. 9): some f_{K_0,eps_0} has beta > 0.58507, so B > 0.58507; Theorem 2.8 (p. 6): beta(f_{K,eps}) = 1/A(K,eps) for every K > 1 and |eps| = 1, with the scaling identity (Lemma 2.2, p. 3), Lemmas 2.3 and 2.4 (p. 4), Proposition 2.5 (p. 5) and the theta-function form (Proposition 2.9 and Corollary 2.10, p. 6) summarized there; Proposition 3.5 (p. 8, proved on p. 9): A_0 < 1.70919, with Lemmas 3.1 (p. 7) and 3.2-3.4 (p. 8), including the certified mesh bound of Lemma 3.4, summarized there. Lemmas 2.1, 2.6 and 2.7 are proof steps, cited on those pages.

Read status. Claims checked: the three results above, with the lemmas named, were read clause by clause on the printed pages, and the short proofs were followed. The ball-arithmetic certification of Lemma 3.4 (Appendix A) was not rerun.

Bears on. #513: Theorem 1.2 gives the lower bound B > 0.58507 for the constant the problem asks for, through the exact formula of Theorem 2.8 and the certified bound of Proposition 3.5. The paper does not determine B and gives no upper bound for it.

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