Differential Algebraic Methods in Multiplicative Number Theory: A Constructive Framework for Explicit Formulae and Prime Distribution

This paper develops a comprehensive differential algebraic framework for multiplicative number theory, establishing explicit representation theorems for multiplicative functions, L-functions, and prime distribution. We construct the multiplicative number theoretic differential closure KMNT through a carefully defined recursive adjunction process that incorporates analytic continuations of constructible families of multiplicative functions, solutions to arithmetic differential equations, and combinatorial correction terms derived from zero distributions and functional equations. Within this closure, we prove that broad classes of multiplicative functions admit explicit representations combining particular solutions with spectral expansions. The framework provides certified error bounds through interval arithmetic and establishes rigorous validation protocols. We develop efficient algorithms with precise complexity analysis and demonstrate applications to prime counting, multiplicative function computation, and connections with random matrix theory and quantum computation. The work bridges differential algebra, multiplicative number theory, and computational mathematics, providing new constructive perspectives on classical problems while maintaining mathematical rigor and practical implementability.