The Regularized Logarithmic Derivative of the Riemann Zeta Function as a Laplace Transform of the Prime-Counting Error

We study the meromorphic function D(s) := -ζ'(s)/sζ(s) - 1/s-1, obtained by removing the polar part of -ζ'(s)/(sζ(s)) at $ s = 1 $. The central observation is the Laplace representation D(s) = ∫₀∞ (ψ(eᵗ) - eᵗ) e⁻ˢᵗ\, dt, Re(s) > 1, which identifies $ D(s) $ as the Laplace transform of the prime-counting error E(t) := ψ(eᵗ) - eᵗ. Six consequences follow, all proved unconditionally: (I) Complete meromorphic structure of D on C: poles at non-trivial zeros ρ (residue -1/ρ), trivial zeros $ -2k $ (residue $ 1/(2k) $), and $ s = 0 $ (residue -log 2π); removable singularity at $ s = 1 $ with D(1) = -1 - γ₀. (II) The Laplace transform converges absolutely for Re(s) > 0 unconditionally and for Re(s) > 1/2 under RH; the abscissa of absolute convergence equals ρ Re(ρ). (III) D(1+i) ∈ L²(R) and 1/2π ∫-∞∞ |D(1+it)|²\, dt = ∫₁∞ (ψ(x)-x)²/x³\, dx < ∞ unconditionally. By contrast, (-ζ'/ζ)(1+i) ∉ L²(R). (IV) Under RH, the right-hand side expands into an explicit, absolutely convergent six-term sum indexed by zeros and constants. (V) Unconditionally, ∫₁∞ (ψ(x)-x)x⁻²\, dx = -1 - γ₀ (Newman's theorem, Theorem 1.7). (VI) D extends holomorphically to Re(s) > 1/2 if and only if the Riemann Hypothesis holds; more precisely, the abscissa of holomorphy of D equals ρ Re(ρ). The paper also includes an independent derivation of ∑ρ Re(1/ρ) = log(2π) - 1 and extends all results to Dirichlet L-functions. Note on corrections. A prior version contained four sign errors: in the residue of D at $ s = 0 $; in the explicit formula for ψ(x) - x; in the zero-sum formula; and in the cross-terms IAL, IAC, ILC of Theorem 1.5. All are corrected here. Recommended Comments field (optional but very useful): 18 pages. Corrected and expanded version (v2). Primary 2020 MSC: 11M06; Secondary: 11N05, 44A10, 11M26. Submitted to Proc. Amer. Math. Soc. Independent researcher.