An exhaustion-aware product moment and zero-free half-planes for Hecke L-functions

Abstract

We refine the ordinary product moment in the cubic-theta comparison by allowing a returned prime product to be exhausted. The return stops either when the positive-prime length inequality holds or when the established prime-free ordinary estimate applies. Keeping the extracted central coefficients gives a local normalization inequality on a larger domain. On the critical ordinary-length interval this yields a net first-mean gain of $1/20000$. We prove the complementary-class, short-sum and algebraic comparison inequalities directly, and obtain the endpoint $B_*-1/160000=0.87446205769353\ldots$, where $B_*$ is Liu's algebraic boundary. The zero-free conclusion is conditional on the cited general-field theta, detector, prior-bound and Mellin input theorems in their stated uniform forms. The proof includes the actual return weights, both approximate-functional-equation arms, the principal-residue normalization and descent to arbitrary number fields.