math.NT — Number Theory
An improved zero-free half-plane for Hecke L-functions over number fields and Quasi-Riemann Hypothesis
In this paper, we obtain a zero-free half-plane Re(s) > B∗ for every finite-order Hecke L- function over every number field, where B∗ = 0.87446830769353045955... is an explicitly defined algebraic number. This improves the Quasi-Riemann Hypothesis over arbitrary num- ber fields. The proof refines the cubic-theta comparison. A sharper amplified Gauss-norm estimate permits a longer additional prime product, chosen separately for each dyadic sum- mation length. Combining the resulting moment saving with contour displacement on the same dyadic sum gives the principal improvement. The enlarged moment controls the critical summation length, where the contour contribution vanishes. Successive improvements of the character-counting estimate then reach the algebraic endpoint, while the numerator-moment parameter remains fixed. We prove optimality within the resulting comparison system. The argument uses the established theta, moment and continuation results of the earlier QRH papers and the general-field formulation of those results.