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.