We develop a residue-based method for recovering the coefficients of aDirichlet series from the singularities of its analytic continuation. Startingfrom the classical vertical-line inversion formula, we introduce a conformaltransformation that converts coefficient extraction into a contour integraland yields a spectral representation in terms of poles and residues. Weapply this framework to the von Mangoldt, prime characteristic, Möbius,and Euler totient functions, obtaining representations involving the zerosand poles of the Riemann zeta function. We also introduce generalized vonMangoldt functions associated with arbitrary Dirichlet series and extendthe approach to finite and Dedekind zeta functions, leading to formulas forpoint counts, prime-ideal counting functions, and characteristic functionsover number fields. Finally, we examine prime-pair, Goldbach, andprime-tuple counting functions and derive conditional error estimatesunder the Riemann hypothesis. These results present contour integrationand zeta-function factorizations as a unified mechanism for translatinganalytic spectral data into arithmetic information.