2024/08/27 by Enderalp Yakaboylu, Yakaboylu, Enderalp
Mathematics · Physics and Astronomy · #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Physics (quant-ph) #Spectral Theory in Mathematical Physics #math-ph #math.MP #quant-ph
paper · pdf · doi:10.48550/arxiv.2408.15135
21 pages. The manuscript has been revised in response to significant comment, ensuring continuous improvement
openalex publication_date 2024/08/27 · openalex created_date 2024/10/25 · arxiv created 2026/07/29 · arxiv updated 2026/07/30 · openalex updated_date 2026/07/31
Let Λ(s) := Γ(s+1) (1-21-s) ζ(s) , and denote its zero set by ZΛ:= Zζ∪ Zp , where Zζ consists of the nontrivial zeros of the Riemann zeta function ζ(s) and Zp of the zeros of the prefactor ( 1-21-s ) , excluding s = 1 . We introduce a non-symmetric operator R on a dense domain D(R) ⊂ L2([0,∞)) with point spectrum σp(R) = \ i(1/2- λ) | λ∈ ZΛ\ . Assuming the simplicity of all nontrivial Riemann zeros, we construct the compression Rζ of R onto the spectral subspace associated with Zζ, and show that Rζ is intertwined with its adjoint by a positive semidefinite operator W ; i.e., W Rζ= Rζ^† W with W ≥ 0 . The positivity of W , viewed as an operator-theoretic form of (Bombieri's refinement of) Weil's positivity criterion, enforces \Re(ρ)=1/2 for all ρ∈ Zζ, in accordance with the Riemann Hypothesis. Under the same positivity condition, the intertwining relation yields a self-adjoint operator whose spectrum coincides with the set \ \Im(ρ) | ρ∈ Zζ\ . We further extend the framework to potential higher-order Riemann zeros and outline its generalization to any Mellin-transformable L -function satisfying a reflection-type functional equation.