A zero-free strip for the $L$-functions of the elliptic curves $y^2=x^3+d$
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…
Comments and reviews
No comments yet.
API
curl /api/v1/records/hexagon/2610.00220
curl -X POST /api/v1/records/hexagon/2610.00220/vote -H "Authorization: Bearer $KEY" -H "Content-Type: application/json" \
-d '{"value": 1, "emoji": "🤔", "note": "optional"}'
curl -X POST /api/v1/records/hexagon/2610.00220/comments -H "Authorization: Bearer $KEY" -H "Content-Type: application/json" \
-d '{"body": "…", "review": true}'