Pair correlations of zeta zeros and perturbations of selfadjoint operators

D. Zaporozhets^ab, V. Kapustin<sup>b

a</sup> Saint Petersburg State University
^b St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences

Abstract: A classical result of H. Montgomery states that the Fourier transform of the pair correlation function for nontrivial zeros of the Riemann zeta function equals $|t|$ for $t\in(-1,1)$. It is known that after a suitable rotation to the real axis, the set of these zeros can be realized as the spectrum of a nonselfadjoint rank-one perturbation of a selfadjoint operator $A$ with “regular” spectrum. Under the assumption that the Riemann hypothesis is true, the rotated set of zeros of the zeta function can be viewed as the spectrum of a selfadjoint perturbation of the same operator $A$, and this perturbation cannot have finite rank. We prove that, moreover, even the weaker Montgomery pair-correlation property cannot hold for any finite-rank perturbation of a regular discrete spectrum. We also show that one can choose a compact perturbation of infinite rank such that its eigenvalues decay faster than any exponential function.

Key words and phrases: Riemann zeta function, Riemann hypothesis, Montgomery's pair correlation conjecture, selfadjoint operators, finite-rank perturbations, compact perturbations, sine kernel determinantal process, Gaussian Unitary Ensemble (GUE), modified Bessel functions, Hilbert–Pólya operator, spectral theory, zeros of the zeta function.

UDC: 511.331, 517.98

Received: 15.11.2025