Topological Phase Transition in the Riemann Zeta Function: A Topological Invariant Characterising the Critical Line

<jats:p>We discover a topological phase transition in the Riemann zeta function. We define a topological invariant ν(α) = Ind(Kα) = (1/2πi)∮_Γ Kα'(z)/Kα(z) dz associated with the coherence kernel Kα. We prove that ν(α) = 0 for α ≠ 1/2 and ν(1/2) = 1. Thus the critical line α = 1/2 is a topological phase transition point: the topological invariant changes discontinuously from ν = 0 to ν = 1 and back. This is a new characterisation of the Riemann Hypothesis: RH is equivalent to the statement that the topological invariant ν(1/2) = 1. The phase transition is analogous to the Anderson localisation-delocalisation transition in disordered systems, the quantum Hall effect, and topological insulators. All results are rigorously proven and fully verifiable numerically.</jats:p>