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_{n≥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(1_B(log N) N^{-σ}) = Σ_{n≥1} n^{-σ} δ_{log n}.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.