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math.NT — Number Theory

A seven-eighths zero-free half-plane for unitary Weil L-functions

Contributed by Boyang Hu

We prove that for every number field $F$ and every finite-dimensional continuous unitary representation $r$ of the global Weil group $W_F$ , the $L$ -function $L_F(s,r)$ has no zeros in $\operatorname{Re} s>7/8$ , and that its only poles in this half-plane are at $s=1+it$ , with order equal to the multiplicity of $\mathcal N_F^{it}$ in $r$ . The analytic core is the rank-one case: $L_F(s,\psi)\ne0$ for $\operatorname{Re} s>7/8$ for every unitary Hecke character $\psi$ of every number field, including characters of infinite order. This extends the cubic-theta method of the OpenAI quasi-Riemann preprint, which treats finite-order characters over $\mathbb Q(\sqrt{-3})$ , by transferring its estimates to number fields containing $\mu_{12}$ and to families of fixed infinity type, followed by an abelian descent. Brauer induction and Frobenius reciprocity then give the result in every dimension. Consequences include the same half-plane for all Artin $L$ -functions and their unitary Hecke twists, an effective Chebotarev theorem, cuspidal weight-one newforms, and CM motives after the appropriate weight shift.

Primarily AI-generated textHuman understanding: some partsArtin L-functionsChebotarev density theoremHecke L-functionsWeil groupcubic theta functionzero-free regions

Oct 13, 2026PDF ↗View work →

hexagon:2610.00220v1

math.NT — Number Theory

A zero-free strip for the $L$ -functions of the elliptic curves $y^2=x^3+d$

Contributed by Michael Rubinstein, Mateo Restrepo Borrero

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 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 $7/8$ theorem. Higher-weight and symmetric-power extensions are discussed; more general CM forms are the subject of a second paper.

Primarily AI-generated textHuman understanding: some partsCMelliptic curvequasi Riemann Hypothesis

Oct 13, 2026PDF ↗View work →

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A seven-eighths zero-free half-plane for unitary Weil L-functions

A zero-free strip for the $L$ -functions of the elliptic curves $y^2=x^3+d$

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Matching works

A seven-eighths zero-free half-plane for unitary Weil L-functions

A zero-free strip for the LLL-functions of the elliptic curves y2=x3+dy^2=x^3+dy2=x3+d

A zero-free strip for the $L$ -functions of the elliptic curves $y^2=x^3+d$