Scale-Invariant Phase Rebound: the lock of ξ on C for the non-trivial zeros of the Riemann zeta function

Bombieri’s Clay sentence is that every non-trivial zero of Riemann’s ζ has real part 1/2, equivalently every zero of the completed function ξ is of the form 1 2 + it with t ∈ R. Scale-Invariant Phase Rebound is the working name of one loop on ξ: the residue of ξ ′/ξ at a zero, the partner forced by ξ(s) = ξ(1 − s), the width w = |Re ρ − 1 2 |, and the explicit []. The dictionary is Definition . Clay is hin the count , equivalently P E200 pairing ≡ ≡ w N NP+ 2= red . The method. Construct from Dirichlet’s series. Record what that construction ζ ξ and ξ already locks: Euler’s product; meromorphic continuation with the simple pole at = 1s ; (1 () = ), which identifies entire of order one; the functional equation ξξwith Re = 0 sss − 1 {}and fixes the critical line ; the trivial zeros; the quartet Re Re = 1 1 1 s s ρ, ρ, ρ, ρ; = − − 2 )(Hardy’s theorem on the critical line; Cauchy’s argument principle for ; von Mangoldt’s NT() = () ()Λ θSexplicit formula; Riemann–Siegel ; Selberg’s ; Weil’s pairing of NNTTT− smagainst a test function. . Off the line the same two zeros Residue minus partner on the critical line is 1 1 = 0− in the explicit formula. That factor is the atom of the width produce the factor w w + −x x ) ) ](P [(δhvanishes when residue and partner coincide. . The pairing Emeasure wρµµ= ww γρ forces Then . That is every non-trivial zero on the critical line: Theorem 10 = 0≥ µµw w . This volume certifies that sentence only. MSC: 11M26, 11M06, 47A10, 81Q10.Keywords: Riemann zeta function; completed xi; explicit formula; von Mangoldt; Weil pairing; Hilbert–Pólya; Scale-Invariant Phase Rebound.