The Generalised Riemann Hypothesis via the Brindel Transformation: Complex Rewriting of n and the ξ-Symmetry of Dirichlet L-Functions

We extend the fixed-point reformulation of the Riemann Hypothesis (Brindel 2026, zenodo.18882529) to all Dirichlet L-functions, establishing the Generalised Riemann Hypothesis (GRH). The key observation is the Brindel transformation: for any n ≥ 2 and k ∈ Z, s_n = 1 + (2πk/ln n)i satisfies n^(s_n) = n. For any primitive Dirichlet character χ, the completed L-function ξ(s,χ) satisfies ξ(s,χ) = ε(χ)·ξ(1-s,χ̄) with |ε(χ)| = 1. Differentiating k times and evaluating at a zero s_0 of order k gives |ξ^(k)(s_0,χ)| = |ξ^(k)(1-s_0,χ)| without any assumption on σ = Re(s_0). At zeros of order k, the Leibniz reduction ξ^(k)(s_0,χ) = G_χ(s_0)·L^(k)(s_0,χ) holds — a non-trivial step specific to zeros. Combined with F_χ(s)·F_χ(1-s) = 1 and |F_χ(s)| = 1 iff σ = 1/2, this implies σ = 1/2 for all non-trivial zeros. Verified numerically at 30 decimal places for χ_3, χ_4, and L(s,Δ) Ramanujan. The Hurwitz zeta function (no functional equation) is tested as a control — its zeros off σ = 1/2 give ratio ≠ 1, confirming the argument is non-trivial.