Hilbert-Pólya Conjecture

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $\zeta(s)$ be the meromorphic continuation of $\sum_{n=1}^{\infty}n^{-s}$, initially defined for $\operatorname{Re}s>1$. A nontrivial zero is a zero $\rho$ with $0<\operatorname{Re}\rho<1$, counted with its analytic multiplicity. A self-adjoint operator $A$ on a complex Hilbert space $\mathcal H$ is a densely defined linear operator whose domain and action equal those of its adjoint. Here discrete spectrum means isolated real eigenvalues with finite-dimensional eigenspaces, with no finite accumulation point.

Statement recorded in the supplied source.

Can one construct a complex Hilbert space $\mathcal H$ and a self-adjoint operator $A$ with purely discrete spectrum such that the multiset of nontrivial zeros of $\zeta$ is exactly \[ \{\tfrac12+i\lambda:\lambda\text{ is an eigenvalue of }A\}, \] where each eigenvalue is repeated according to the dimension of its eigenspace? [S2]

Short English statement

Can the nontrivial zeta zeros be realized, with their multiplicities, by the real eigenvalues of a self-adjoint operator after shifting by one-half and multiplying by the imaginary unit?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1275 (NT-068). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Sebastian Endres and Frank Steiner, The Berry--Keating operator on $L^2(\mathbb R_{>},dx)$ and on compact quantum graphs with general self-adjoint realizations, arXiv:0912.3183v5 (2011), Section 1, pp. 2--3. https://arxiv.org/pdf/0912.3183.