On Erd˝ os's power-sum problem #973: lim sup m_n ≤ 0.0773

For n ≥ 2 let m_n be the infimum of max_{2≤k≤n+1} |z_1^k +··· + z_n^k| over all z_1 = 1 and |z_i| ≥ 1. Erd˝ os asked whether m_n < C^(−n) for a constant C > 1 (Problem #973 in Bloom's list, Problem 7.3 in Hayman's list). Luo, Yang and Zhu answered this in the negative, and Tan, Wang, Huang and Chen proved m_n ≥ exp(−(1+o(1))√n log n); whether m_n → 0 was left open, and the only upper bound recorded in the literature was m_n ≤ n. We prove lim sup_{n→∞} m_n ≤ 0.0773. The configurations are the reciprocals of the zeros of Taylor sections of e^(Bw)(1−w)^ε, whose power sums of orders 2, ..., n+1 all equal ε once the coefficient of w^(n+1) vanishes; the difficulty is to show that all zeros lie in the unit disc. We prove a local limit theorem near w = 1, in which a truncated Laplace transform of the parabolic cylinder function D_ε appears, control the rest of the circle by two Abel summations, and certify the needed properties of the limit model in ball arithmetic with two independent programs. We also certify m_4000 ≤ 0.08163 and bounds for smaller n, and show that on the branch used in the proof the construction fails for ε = 0.0772. We conjecture lim m_n = 0.07725...; in particular, m_n would not tend to 0.  Source code and certificates are attached: this record contains the manuscript (PDF) and supplementary_code.zip with all source code, certificates, logs, SHA-256 hashes and a README that maps every statement of the paper to the program that verifies it.