Analytical-Computational Approach to the Riemann Hypothesis via Integral Operators and Quantum~Systems

This work presents a new approach to investigate the Riemann Hypothesis, combining analytical and computational methods. We develop a procedure to construct an integral operator K from the Fourier analysis of the prime counting error E(x) = π(x) −li(x). We investigate the hypothesis that the eigenvalues of this operator correspond to the imaginary parts of the non-trivial zeros of the Riemann zeta function ζ(s). Analytically, we examine the consequences of a possible normalization of the form ζ(s) = (1/π)arcsinh(Z(s)) +1/2, where Z(s) is a meromorphic function. We show that this structure imposes strong constraints on the location of zeros in the complex plane. Computationally, weverify our construction for the first 2000 zeros, obtaining correspondence with precision of 10−12. The statistical distribution of eigenvalue spacings follows the Gaussian Unitary Ensemble (GUE) with a p-value of 0.3129, consistent with known properties of the zeros of ζ(s). This study suggests new connections between analytic number theory, spectral theory of operators, and quantum systems, offering a promising perspective for future investigations of the Riemann Hypothesis.