Nontrivial Riemann Zeros as Spectrum

Abstract

Let $ \Lambda(s) := \Gamma(s+1)\, (1-2^{1-s}) \, \zeta(s) $, and write its zero set as $ \mathcal{Z}_\Lambda := \mathcal{Z}_\zeta \cup \mathcal{Z}_\mathrm{p} $, where $ \mathcal{Z}_\zeta $ collects the nontrivial zeros of the Riemann zeta function $ \zeta(s) $ and $ \mathcal{Z}_\mathrm{p} $ the zeros of the prefactor $ (1-2^{1-s}) $ other than $ s = 1 $. We introduce a densely defined non-symmetric operator on the half-line, $ \hat{\mathcal{R}} \colon \mathcal{D}(\hat{\mathcal{R}}) \subset L^2([0,\infty)) \to L^2([0,\infty)) $, and solve the eigenvalue equations of $ \hat{\mathcal{R}} $ and its adjoint $ \hat{\mathcal{R}}^\dagger $ in closed form: the point spectra are \[ \sigma_{\mathrm{p}}(\hat{\mathcal{R}}) = \left\{ i\left(1/2- \lambda \right) \mid \lambda \in \mathcal{Z}_\Lambda \right\} , \qquad \sigma_{\mathrm{p}}(\hat{\mathcal{R}}^\dagger) = \overline{\sigma_{\mathrm{p}}(\hat{\mathcal{R}})} \, , \] and their eigenstates form a biorthogonal system. Assuming that all nontrivial Riemann zeros are simple, this biorthogonal structure furnishes compressions $ \hat{\mathcal{R}}_\zeta $ and $ \hat{\mathcal{R}}_\zeta^\dagger $ of $ \hat{\mathcal{R}} $ and $ \hat{\mathcal{R}}^\dagger $ to their respective spectral subspaces associated with $ \mathcal{Z}_\zeta $. We prove that these compressions are intertwined by a symmetric operator $ \hat{W} $---that is, $ \hat{W}\,\hat{\mathcal{R}}_\zeta = \hat{\mathcal{R}}_\zeta^\dagger\,\hat{W} $---whose positivity is equivalent to the Riemann Hypothesis. This positivity condition is an operator-theoretic form of the Weil--Bombieri positivity criterion, and entails the existence of a self-adjoint Hilbert--P\'olya operator. We further extend the framework to potential higher-order Riemann zeros and outline its generalization to Mellin-transformable $ L $-functions satisfying a reflection-type functional equation.