Angular Monotonicity of Completed Dirichlet L-Functions via the Three-Term Zeta Decomposition: A Proof of the Generalized Riemann Hypothesis for Non-Principal Characters

Abstract

Version 6, 2026-08-31. This version adds links to the companion interactive web pages (English/Chinese expository sites) and the GitHub code repository. No mathematical content changes from v5. We prove the Generalized Riemann Hypothesis (GRH) for all non-principal Dirichlet L-functions L(s,chi) with conductor q >= 3. The proof proceeds by combining the three-term polylogarithm decomposition of the Riemann zeta function with the completed L-function Lambda(s,chi) to form a product xi_K(s) = xi(s) * Lambda(s,chi). We then establish angular monotonicity of xi_K in polar coordinates centered at s = 1/2, which forces all zeros of xi_K (and hence all zeros of L(s,chi)) to lie on the critical line Re(s) = 1/2. The key ingredient is the non-negativity of the coefficients c_n = 1 - z_1^n - z_2^n, where z_1 = tan(pi/8), z_2 = 1 - tan(pi/8), which follows from elementary inequalities without assuming GRH. This establishes a direct bridge between the arithmetic positivity of the Dirichlet coefficients and the analytic monotonicity required for the zero-free region.