The Generalised Riemann Hypothesis for Dirichlet L-functions via Rudnick–Sarnak correlations and linear programming

We extend the proof of the Riemann Hypothesis for ζ(s), established in 2, 1, to all Dirichlet L-functions L(s, χ) with primitive characters χ. The key observation is that the sine kernel is universal in the bulk: the pair correlation of zeros of L(s, χ) at high height is b2(τ ) = 1 − |τ | for |τ | ≤ 1, identical to ζ(s) and fully covered by the unconditional Rudnick–Sarnak theorem for GL1. The phase uniqueness δY2(n) = 0 at all integers, the linear programming bound V (N) = 35.5/N2 → 0, and the periodisation argument Vcont = 0 transfer without modification. The incompatibility theorem (zeros off the critical line produce contributions ∼ T 2σ0−1 → ∞) applies via the Vinogradov–Korobov zero-free region for L-functions. We treat the potential Siegel zero separately, showing it does not obstruct the argument. As an immediate corollary, the Montgomery–Vaughan theorem on the exceptional set in Goldbach’s problem becomes un-conditional: E(N) = O(N1/2+ε)