Logarithmic Dirac Combs and Spectral-Measure Realization for the Riemann Zeta Function

This paper develops a unified arithmetic framework for the Riemann zeta function from two complementary realizations of the same integer data. First, it formulates the weighted logarithmic Dirac comb as a finite discrete Borel measure obtained as the logarithmic pushforward of the weighted arithmetic measure on (0, ∞). This gives a measure-theoretic realization of multiplicative arithmetic in additive logarithmic variables. Second, it introduces the discrete Hilbert space ℓ² (ℕ), the completeness of the integer-labeled projections, and the integer-spectrum operatorN = Σ₍≥₁ n |n⟩⟨n|, whose logarithm carries the logarithmic arithmetic spectrum. The central theorem identifies the weighted logarithmic Dirac comb exactly with a weighted spectral measure of log N: μ_σ (B) = Tr (1B (log N) N^-σ) = Σ₍≥₁ n^-σ δ₋₎₆ ₍. Equivalently, ∫_ℝ f (x) dμ_σ (x) = Tr (f (log N) N^-σ). This provides a precise bridge between logarithmic arithmetic measures and spectral operator data. As consequences, the paper recovers Tr (N^-s) = ζ (s), the Fourier/Mellin realization of the weighted logarithmic comb, the Euler product in prime-mode form, and the von Mangoldt weighted trace formula for −ζ′ (s) /ζ (s). The paper does not address the nontrivial zeros or claim a solution of the Riemann hypothesis.