Nontrivial Riemann Zeros as Spectrum

Abstract

Let $ \Lambda (s) := \Gamma(s+1)\, (1-2^{1-s}) \, \zeta(s) $, and denote its set of zeros by $ Z_\Lambda := Z_\zeta \cup Z_\mathrm{p} $, where $ Z_\zeta $ consists of the nontrivial zeros of $ \zeta(s) $ and $ Z_\mathrm{p} $ those of the prefactor $ ( 1-2^{1-s} ) $, with $ s \neq 1 $. We introduce a non-symmetric operator $ R $ on $ L^2([0,\infty)) $ with spectrum \[ \sigma(R) = \left\{ i\left(1/2- \lambda \right) \mid \lambda \in Z_\Lambda \right\} \, . \] Assuming the simplicity of all nontrivial Riemann zeros, we construct the compression $ R_{Z_\zeta} $ of $ R $ to the spectral subspace associated with $ Z_\zeta$, and show that $ R_{Z_\zeta} $ is intertwined with its adjoint by a positive semidefinite operator $ W $; i.e., $ W \, R_{Z_\zeta} = R_{Z_\zeta}^\dagger \, W $ with $ W \ge 0 $. The positivity of $ W $, viewed as an operator-theoretic form of (Bombieri's refinement of) Weil's positivity criterion, enforces $ \Re(\rho)=1/2 $ for all $ \rho \in Z_\zeta $, 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(\rho) \mid \rho \in Z_\zeta\} $. We further extend the framework to accommodate higher-order nontrivial Riemann zeros, should they exist, and to cover any Mellin-transformable $ L $-function satisfying a functional equation.