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, $\hat{\mathcal{R}}:\mathcal{D}(\hat{\mathcal{R}})\subset L^2([0,\infty ))\to L^2([0,\infty ))$, and solve the eigenvalue equations of $\hat{\mathcal{R}}$ and its adjoint ${\hat{\mathcal{R}}}^{†}$ in closed form: the point spectra are \[ σ_{\mathrm{p}}(\hat{\mathcal{R}}) = \left\{ i\left(1/2- λ\right) \mid λ\in \mathcal{Z}_Λ\right\} , \qquad σ_{\mathrm{p}}(\hat{\mathcal{R}}^\dagger) = \overline{σ_{\mathrm{p}}(\hat{\mathcal{R}})} \, , \] 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 ${\hat{\mathcal{R}}}_{\zeta }^{†}$ of $\hat{\mathcal{R}}$ and ${\hat{\mathcal{R}}}^{†}$ 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, $\hat{W}{\hat{\mathcal{R}}}_{\zeta }={\hat{\mathcal{R}}}_{\zeta }^{†}\hat{W}$---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--Pólya 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.

19 pages. The manuscript has been revised in response to significant comment, ensuring continuous improvement