Nontrivial Riemann Zeros as Spectrum

Let $\Lambda (s) := \Gamma(s+1)\, (1-2^{1-s}) \, \zeta(s)$ , and denote its set of zeros by $\mathcal{Z}_\Lambda := \mathcal{Z}_\zeta \cup \mathcal{Z}_\mathrm{p}$ , where $\mathcal{Z}_\zeta$ consists of the nontrivial zeros of $\zeta(s)$ and $\mathcal{Z}_\mathrm{p}$ those of the prefactor $( 1-2^{1-s} )$ , with $s \neq 1$ . We introduce a non-symmetric operator $\hat{\mathcal{R}}$ on $L^2([0,\infty))$ with spectrum

$\displaystyle \sigma(\hat{\mathcal{R}}) = \left\{ i\left(1/2- \lambda \right) \mid \lambda \in \mathcal{Z}_\Lambda \right\} \, .$

Assuming simplicity of all nontrivial Riemann zeros, we construct the compression $\hat{\mathcal{R}}_{\mathcal{Z}_\zeta}$ of $\hat{\mathcal{R}}$ to the spectral subspace associated with $\mathcal{Z}_\zeta$ , and show that $\hat{\mathcal{R}}_{\mathcal{Z}_\zeta}$ is intertwined with its adjoint by a positive semidefinite operator $\hat{W}$ ; i.e., $\hat{W} \, \hat{\mathcal{R}}_{\mathcal{Z}_\zeta} = \hat{\mathcal{R}}_{\mathcal{Z}_\zeta}^\dagger \, \hat{W}$ with $\hat{W} \ge 0$ . The positivity of $\hat{W}$ , viewed as an operator-theoretic form of (Bombieri’s refinement of) Weil’s positivity criterion, enforces $\Re(\rho)=1/2$ for all $\rho \in \mathcal{Z}_\zeta$ , 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 $\{ \Im(\rho) \mid \rho \in \mathcal{Z}_\zeta\}$ . 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.

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Nontrivial Riemann Zeros as Spectrum