What question did this study set out to answer?

The aim is to explore the spectral properties of Riemann zeros using a novel operator on a prime number lattice.

April 3, 2026Open Access

Prime Number Lattice Operator and Spectral Invariants: A Fredholm Regularization of the Riemann Zeta Function

Key Points

The aim is to explore the spectral properties of Riemann zeros using a novel operator on a prime number lattice.
Introduced a self-adjoint operator associated with the prime number lattice.
Analyzed spectral statistics adhering to the Gaussian Unitary Ensemble.
Calculated the Fredholm determinant as a function of the operator.
Identified the invariant scattering phase on the critical line.
Achieved a KS p-value of 0.905 for spectral statistics following GUE.
Found a stationary scattering phase approximately equal to 0.345.
Demonstrated a non-vanishing spectral gap Δ∞ approximately equal to 0.961.

Abstract

We present Version 6. 0 of the spectral realization of Riemann zeros on the prime numberlattice L = log p: p ∈ P. This paper introduces a self-adjoint operator ˜K whose spectralstatistics strictly follow the Gaussian Unitary Ensemble (GUE) with a KS p-value of 0. 905. We demonstrate that the Fredholm determinant det (I − e−s ˜K) acts as a spectral detectorfor the non-trivial zeros γn. A fundamental discovery in this version is the existence of aninvariant scattering phase Φ (s) = Re log det (I−e−s ˜K) ≈ 0. 345, which remains stationaryon the critical line with sharp resonances at t = γn. We prove that ˜K effects a Fredholmregularization of the Hadamard product for ζ (s). Numerical evidence for a non-vanishingspectral gap Δ∞ ≈ 0. 961 is provided, suggesting a structural link to the Yang-Mills massgap analogy.

Prime Number Lattice Operator and Spectral Invariants: A Fredholm Regularization of the Riemann Zeta Function

Key Points

Abstract

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