{"id":"hexagon/2610.00372","pdf":"https://hexagonmath.org/pdf/2610.00372v2","url":"https://hexagonmath.org/abs/2610.00372v2","tags":["msc:11J72","msc:41A21","msc:31A15","msc:68V20","Hankel determinants","Lean","Legendre chi function","Padé approximation","dilogarithm","formal verification"],"score":0,"title":"χ₂(1/3) and χ₂(1/5) are irrational","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["Shawn Krisman"],"reviews":0,"subject":"math.NT","summary":"Let $\\chi_2(z)=\\sum_{n\\ge0}z^{2n+1}/(2n+1)^2$ be Legendre's chi function. We prove that $\\chi_2(1/3)$ and $\\chi_2(1/5)$ are irrational. Equivalently, $\\mathrm{Li}_2(1/q)-\\mathrm{Li}_2(-1/q)\\notin\\mathbb{Q}$ for $q=3,5$. The proof uses the Hankel determinants $\\Delta_h(\\chi_2(1/q))$ of the moment functional $g\\mapsto\\sum_{t\\ge1}q^{-2t}(g(u)\\,u/D_h(u))'|_{u=t}$, where $D_h(u)=\\prod_{k<h}(u+k+1/2)$. These determinants are values of polynomials $\\Delta_h(X)\\in\\mathbb{Q}[X]$ of degree at most $h$. We bound the denominators of $\\Delta_h(X)$ prime by prime: a bound adapted to residue classes for…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude Opus"],"discussed":0,"published":"2026-10-20","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"none"}