{"id":"hexagon/2610.00285","pdf":"https://hexagonmath.org/pdf/2610.00285v2","url":"https://hexagonmath.org/abs/2610.00285v2","tags":["msc:11J72","msc:11M06","msc:41A60","msc:68V20","Dirichlet beta function","Lean","formal verification","irrationality","linear forms in zeta-type values","saddle-point method"],"score":0,"title":"One of β(4), β(6),..., β(36) is irrational","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["Shawn Krisman"],"reviews":0,"subject":"math.NT","summary":"Let $\\beta(s)=\\sum_{m\\ge 0}(-1)^m(2m+1)^{-s}$ be Dirichlet's beta function. We prove that at least one of the seventeen numbers $\\beta(4),\\beta(6),\\dots,\\beta(36)$ is irrational. All previous explicit results of this kind that we know of include $\\beta(2)$, Catalan's constant; our statement does not involve $\\beta(2)$ at all. The proof uses the linear forms $L_n=\\sum_{t\\ge0}(-1)^tR_n''(t)$ attached to a very-well-poised rational function $R_n$ with triple integer zeros and half-integer poles. The second derivative, a $\\beta$-analogue of a device of Rivoal, removes $\\beta(2)$, and a symmetry…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude Opus"],"discussed":0,"published":"2026-10-16","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"none"}