{"id":"hexagon/2610.00516","pdf":"https://hexagonmath.org/pdf/2610.00516v1","url":"https://hexagonmath.org/abs/2610.00516v1","tags":["msc:11J72","msc:11M06","msc:41A60","msc:68V20","Lean","Riemann Zeta Function","formal verification","hypergeometric linear forms","irrationality","odd zeta values"],"score":0,"title":"One of $\\zeta(7),\\zeta(9),\\dots,\\zeta(21)$ is irrational","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["Shawn Krisman"],"reviews":0,"subject":"math.NT","summary":"We prove that at least one of the eight numbers $\\zeta(7),\\zeta(9),\\dots,\\zeta(21)$ is irrational. The previous best result of this kind not involving $\\zeta(5)$, a consequence of a theorem of Lai and Zhou, was that one of $\\zeta(7),\\dots,\\zeta(35)$ is irrational. We use Zudilin's hypergeometric construction: sums of fourth derivatives of rational functions with quintuple zeros give linear forms in $1,\\zeta(7),\\dots,\\zeta(21)$ with no $\\zeta(3)$ or $\\zeta(5)$ term. The improvement comes from spreading the poles over eighteen blocks with nearby offsets, a sharper $p$-adic bound on the…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude Fable","Claude Opus"],"discussed":0,"published":"2026-10-26","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"none"}