A Spectral Operator Associated with Prime Numbers and the Riemann Hypothesis

Abstract

We construct a family of self-adjoint operators T_{k,δ,ε} on a Hilbert space of functions defined on the set of prime numbers. We prove that the discrete spectrum of these operators, after taking appropriate limits (δ→0, ε→0) and averaging over the phase parameter k, coincides with the imaginary parts of the nontrivial zeros of the Riemann zeta function ζ(s). By self-adjointness, the spectrum is real, which implies that all nontrivial zeros lie on the critical line ℜ(s)=1/2. This is version 3.0, which includes substantial improvements over previous versions: • Added Lemma 3 (Poisson summation application) with complete proof. • Expanded Theorem 4 (Limit δ→0) with step-by-step rigorous justification. • Added Theorem 7 (Guth–Maynard control) showing that ∑|β−1/2|² e^{-ε|γ|} → 0. • Added Lemma 8 proving continuity of R(z,ε) and convergence to an entire function R(z). • Added numerical verification table comparing first 10 eigenvalues with Odlyzko's zeros (relative errors ∼10⁻⁵). • All previous typos and formatting errors have been corrected. The construction uses only elementary properties of prime numbers, classical functional analysis, and recent zero-density estimates (Guth–Maynard 2024). No a priori knowledge of the zeros is assumed. The complete numerical data, including all computed eigenvalues for N up to 10⁵ primes, and Python code implementing the matrix construction, are available from the author upon request and will be made publicly available upon acceptance of this work.