HEXAGONbeta

A zero-free strip for the LL-functions of the elliptic curves y2=x3+dy^2=x^3+d

Primarily AI-generated textHuman understanding: some partsmath.NT — Number Theory

Contributed by Michael Rubinstein ↗, Mateo Restrepo Borrero

Version 1 / Oct 13, 2026 / CC BY 4.0

Abstract

For a nonzero integer dd let EdE_d be the elliptic curve y2=x3+dy^2=x^3+d over Q\mathbb Q. Part I adapts OpenAI's 11/1211/12 argument to prove that L(Ed,s)≠0L(E_d,s)\ne0 for ℜs>17/12\Re s>17/12. Part II adapts OpenAI's 7/87/8 argument to prove L(s,α‾η)≠0L(s,\overline\alpha\eta)\ne0 for ℜs>7/8\Re s>7/8, where η\eta is any finite-order Hecke character of Q(−3)\mathbb Q(\sqrt{-3}) and α(n)=n/∣n∣\alpha(n)=n/|n| on primary generators. Consequently L(Ed,s)≠0L(E_d,s)\ne0 for ℜs>11/8\Re s>11/8. The analytic normalization is L(Ed,s+12)L(E_d,s+\frac12), so the two boundaries are 11/1211/12 and 7/87/8 within 0<ℜs<10<\Re s<1. The curve characters have infinite order, and require new transformation and moment arguments. These are proved here; steps identical to the original arguments are cited. Part II also uses the ideas from OpenAI's finite-order 7/87/8 theorem. Higher-weight and symmetric-power extensions are discussed; more general CM forms are the subject of a second paper.

Provenance statement

This paper adapts OpenAI’s 11/12 and 7/8 arguments to the L-functions of the elliptic curves y²=x³+d. Much of the mathematical development, writing, and test code was produced using Claude Opus 5.5 and OpenAI Astra/Codex. We do not claim that the complete proof has been independently checked and understood by a human mathematician. In particular, the new CM transformation arguments, uniform moment estimates, descent, and 7/8 adaptation have been developed and audited primarily by AI. A lean certification of the proofs in the paper is being implemented. In the meantime, Python, PARI/GP, SymPy, and mpmath were used for exact, symbolic, and numerical checks. Section 18 describes their scope and limitations. These checks test finite instances and normalizations; they do not prove the uniform analytic estimates.

Tools used

Anthropic
Claude OpusVersion 5.5
OpenAI
CodexVersion Astra

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Version history

  1. v1Submitted by Michael RubinsteinInitial depositCurrentOct 13, 2026