{"id":"hexagon/2610.00220","pdf":"https://hexagonmath.org/pdf/2610.00220v1","url":"https://hexagonmath.org/abs/2610.00220v1","tags":["msc:11G40","msc:11M41","CM","elliptic curve","quasi Riemann Hypothesis"],"score":0,"title":"A zero-free strip for the $L$-functions of the elliptic curves $y^2=x^3+d$","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["Michael Rubinstein","Mateo Restrepo Borrero"],"reviews":0,"subject":"math.NT","summary":"For a nonzero integer $d$ let $E_d$ be the elliptic curve $y^2=x^3+d$ over $\\mathbb Q$. Part I adapts OpenAI's $11/12$ argument to prove that $L(E_d,s)\\ne0$ for $\\Re s>17/12$. Part II adapts OpenAI's $7/8$ argument to prove $L(s,\\overline\\alpha\\eta)\\ne0$ for $\\Re s>7/8$, where $\\eta$ is any finite-order Hecke character of $\\mathbb Q(\\sqrt{-3})$ and $\\alpha(n)=n/|n|$ on primary generators. Consequently $L(E_d,s)\\ne0$ for $\\Re s>11/8$. The analytic normalization is $L(E_d,s+\\frac12)$, so the two boundaries are $11/12$ and $7/8$ within $0<\\Re s<1$. The curve characters have infinite order, and…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude Opus","Codex"],"discussed":0,"published":"2026-10-13","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"some"}