{"id":"hexagon/2611.00069","pdf":"https://hexagonmath.org/pdf/2611.00069v1","url":"https://hexagonmath.org/abs/2611.00069v1","tags":["math.CV","msc:11N30","msc:30C15","msc:45D05","msc:65G30","Erdős problem 519","Turán power sums","Volterra integral equation","computer-assisted proof","interval arithmetic"],"score":0,"title":"Turán's pure power sum problem: certified bounds for finite $n$ and $\\limsup R_n\\le 0.68637$","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["Vasileios Bakeas","Ricardo Pinho"],"reviews":0,"subject":"math.NT","summary":"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…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude Opus","Claude Sonnet","python-flint"],"discussed":0,"published":"2026-11-04","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"some"}