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MP: Fachverband Theoretische und Mathematische Grundlagen der Physik
MP 5: Quantum Mechanics: Spectral Theory and Many-Body Systems
MP 5.1: Hauptvortrag
Mittwoch, 18. März 2026, 13:45–14:15, KH 02.013
Nontrivial Riemann Zeros as Spectrum — •Enderalp Yakaboylu — Department of Physics and Materials Science, University of Luxembourg, L-1511 Luxembourg
Define Υ(s) := Γ(s+1) (1−21−s) ζ(s) , and denote by Z := {γ ∈ C | Υ (γ)=0 } its set of zeros, which includes both the periodic eta zeros, determined by (1−21−s) =0 with s ≠ 1 , and the nontrivial zeta zeros ρ. We introduce a non-symmetric operator
| R ∶ D(R) ⊂ L2([0,∞)) → L2([0,∞)) , |
with spectrum
| σ(R) = | ⎧ ⎨ ⎩ | i | ⎛ ⎝ | 1/2− γ | ⎞ ⎠ | | γ ∈ Z | ⎫ ⎬ ⎭ | . |
Assuming that all nontrivial zeros of the zeta function are simple, we construct a positive semidefinite operator Ŵ intertwining R and its adjoint on the spectral subspace associated with the nontrivial zeros,
| R†Ŵ = Ŵ R . |
The positivity of Ŵ , which represents an operator-theoretic form of (Bombieri’s refinement of) Weil’s positivity criterion, enforces ℜ(ρ)=1/2 for all ρ , in accordance with the Riemann Hypothesis. Furthermore, from the similarity between R and R†, we obtain a self-adjoint Hilbert-Pólya operator, whose spectrum coincides with the imaginary parts of the nontrivial zeta zeros.
The presented framework can be generalized to higher-order zeta zeros, if such exist, and to any other Mellin-transformable L-function satisfying a functional equation.
Keywords: Hilbert-Pólya Conjecture; Berry-Keating Hamiltonian; Riemann Zeta Function