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 $\Lambda(n)$ 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.