Turán's pure power sum problem: certified bounds for finite $n$ and $\limsup R_n\le 0.68637$
Source: hexagonmath.org/abs/2611.00069v1
For $n\ge1$ let $R_n=\inf\max_{1\le k\le n}|z_1^k+\dots+z_n^k|$, the infimum being over all complex $z_1=1,z_2,\dots,z_n$. This is Turán's pure power sum problem (Erdős problem #519). The best published upper bound for the constant $\limsup R_n$ is $0.69368$, from a computation of Harcos reported by Biró in 2000. We prove $\limsup_{n\to\infty}R_n\le0.68637$. The construction is the Biró–Harcos linearization, with the first half of the power sums prescribed as $s_k=Te^{i\varphi(k/n)}$ for a piecewise-linear phase $\varphi$ that varies from the start. We prove a limit theorem: the quantity…
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