A Novel Higher-Order Framework Connecting Prime Zeta Functions to Dirichlet Convolutions and Prime Decomposition

This paper introduces a novel higher-order analytical framework connecting prime zeta functions to Dirichlet convolutions and prime decomposition, extending the foundational non-recursive partial fraction framework established by Saif R. Lazim. By utilizing higher-order derivatives and generalized arithmetic expansions, we derive rigorous bridge identities for the von Mangoldt function Λ(n) and establish generalized Euler product representations. These results provide an advanced mathematical toolkit for analyzing the distribution of prime numbers, prime powers, and associated arithmetic series.