Spectral Representation of the Zeros of the Riemann Zeta Function: An Experimental Mathematics Study

This work investigates the construction of an integral operator K∞ acting in the space L2(μ∞), where μ∞ is a measure on the interval (0, 1] generated by the limit points of the reciprocals of prime numbers. A systematic numerical study of the spectrum of finite-dimensional approximations KN of the operator K∞ is carried out depending on the choice of metric on the set of primes. It is found that all studied metrics (|1/p − 1/q|, |√p −√q|, | log p − log q|, Gaussian kernel |p −q|) demonstrate high correlation (0.93–0.999) between the normalized eigenvalues γK,n = −log(λn/λ1) and the imaginary parts γζ,n of the nontrivial zeros of the Riemann zeta function. The highest correlation (0.9995) is achieved for the metric √p −√q|, which practically corresponds to an ideal linear relationship. For all metrics, a systematic dependence γζ,n = α(N)γK,n +β(N)+Rn is observed, where the slope coefficient α(N) → 1 as N → ∞ according to the law 1 − c/ logN with c ≈ 36.2, and β(N) ≈ −logN. The residuals Rn exhibit an oscillatory structure with amplitude decreasing as n increases. The self-adjointness of the operator K∞ implies the reality of all γn, which is equivalent to the Riemann Hypothesis. This work represents an example of experimental mathematics, where numerical observation precedes and motivates rigorous theory.