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An Ergodic Hypothesis for Logarithms of Primes and its Equivalence to the Montgomery Pair Correlation Conjecture
We formulate a precise ergodic hypothesis concerning the uniform pairwise distribution of the weighted set {log p : p prime} on the real line. The hypothesis states that for any smooth compactly supported function f,
lim_{X to infinity} 1/(log X)^2 sum_{p,q le X} (log p)^2/p (log q)^2/q f(log p - log q) = int_{-infinity}^{infinity} f(t) dt.
We construct a family of finite-dimensional Hermitian operators H_{X,N}, built from prime numbers, whose eigenvalues are explicit trigonometric sums involving log p. Through a detailed autocorrelation analysis of these eigenvalues employing the discrete Fourier transform, the method of moments, and meticulous error control, we prove that the ergodic hypothesis is strictly equivalent to the Montgomery pair correlation conjecture for the zeros of the Riemann zeta function. The equivalence is established unconditionally and provides a new, directly verifiable formulation of one of the central open problems in analytic number theory. All estimates are fully rigorous, and the operator construction is elementary, requiring only finite-dimensional linear algebra and classical harmonic analysis.
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