Hilbert–Pólya conjecture

Let $\zeta$ denote the Riemann zeta function, that is, the meromorphic continuation to $\mathbb{C}$ of the function $\zeta(s)=\sum_{n=1}^{\infty}n^{-s}$, $\operatorname{Re} s>1$. Call $\rho\in\mathbb{C}$ a non-trivial zero of $\zeta$ if $\zeta(\rho)=0$ and $0<\operatorname{Re}\rho<1$, and let $Z$ be the multiset of non-trivial zeros, each zero counted with its multiplicity as a zero of $\zeta$. For $\rho\in Z$ define

$$t_{\rho}=-i\left(\rho-\tfrac{1}{2}\right)\in\mathbb{C},\qquad\text{so that}\qquad \rho=\tfrac{1}{2}+it_{\rho}.$$

There exist a separable complex Hilbert space $\mathcal{H}$, a dense linear subspace $D\subseteq\mathcal{H}$, and a self-adjoint (possibly unbounded) operator $H\colon D\to\mathcal{H}$ such that

$$\operatorname{spec}(H)=\left\{\,t_{\rho}\;:\;\rho\in Z\,\right\},$$

the spectrum consisting purely of eigenvalues, and such that for every $t\in\operatorname{spec}(H)$ the dimension of the eigenspace $\ker(H-t\,\mathrm{id})$ equals the multiplicity of $\tfrac{1}{2}+it$ as a zero of $\zeta$.

Equivalently, in terms of the operator $L=\tfrac{1}{2}+iH$ defined on $D$: $\operatorname{spec}(L)=Z$ with matching multiplicities, where $H=-i\left(L-\tfrac{1}{2}\right)$ is self-adjoint on $\mathcal{H}$.