{"id":"hexagon/2610.00714","pdf":"https://hexagonmath.org/pdf/2610.00714v1","url":"https://hexagonmath.org/abs/2610.00714v1","tags":["cs.LO","msc:11M26","msc:11M20","msc:11M06","msc:68V20","msc:03B35","Dirichlet L-functions","Gauss sums","Lean 4","Mathlib"],"score":0,"title":"Every real primitive Dirichlet L-function has de Bruijn–Newman constant at most 9/32: a kernel-checked proof in Lean 4","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["Peter W. Murphy, MD"],"reviews":0,"subject":"math.NT","summary":"Let χ be a real primitive Dirichlet character of conductor N ≥ 3 and parity k ∈ {0,1}. Let Λ(s,χ) = (N/π)^{(s+k)/2} Γ((s+k)/2) L(s,χ) be its completed L-function, and let H^χ_t(z) = ∫_0^∞ e^{tu²} Φ_χ(u) cos(zu) du be the de Bruijn–Newman heat flow of Λ(·,χ), normalised so that H^χ_0(z) = (1/8) Λ(1/2 + iz/2, χ). We give a proof, checked by the Lean 4 kernel, that for every such χ and every t ≥ 9/32 all zeros of H^χ_t are real. Equivalently, the generalised de Bruijn–Newman constant satisfies Λ_χ ≤ 9/32. The mathematics is not new, and we say exactly whose it is. De Bruijn's strip-contraction…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude"],"discussed":0,"published":"2026-11-02","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"some"}