The quasi-Riemann hypothesis: a fixed zero-free half-plane for zeta and all Dirichlet L-functions

Theorem 1 claims that every finite-order Hecke $L$-function over $\mathbb Q(\sqrt{-3})$ and every Dirichlet $L$-function, including $\zeta(s)$, has no zero in $\Re s>7/8$, the principal pole at $s=1$ excepted. So the nontrivial zeros of $\zeta$ have real part at most $7/8$, and any exceptional real zero in $(7/8,1)$ is excluded. The proof runs through cubic theta series and sextic large sieves, first reaching $11/12$ and then refining to $7/8$; the Dirichlet case is transferred from the Hecke family. It does not prove the Riemann hypothesis or GRH, and says so; the boundary line $\Re s=7/8$ is not covered. A separate manuscript gives a different proof of the weaker half-plane $\Re s>11/12$.

Theorem 1 claims that every finite-order Hecke $L$-function over $\mathbb Q(\sqrt{-3})$ and every Dirichlet $L$-function, including $\zeta(s)$, has no zero in $\Re s>7/8$, the principal pole at $s=1$ excepted. So the nontrivial zeros of $\zeta$ have real part at most $7/8$, and any exceptional real zero in $(7/8,1)$ is excluded. The proof runs through cubic theta series and sextic large sieves, first reaching $11/12$ and then refining to $7/8$; the Dirichlet case is transferred from the Hecke family. It does not prove the Riemann hypothesis or GRH, and says so; the boundary line $\Re s=7/8$ is not covered. A separate manuscript gives a different proof of the weaker half-plane $\Re s>11/12$.