Multibranch Extensions of the Completed Riemann Zeta Function

This work introduces a branch-indexed generalization of the completed Riemann zeta function, developed through a detailed analysis of the multivalued nature of complex exponentiation. By constructing a countable family of modified zeta and completed zeta functions, each corresponding to a distinct branch of the complex logarithm, we examine how analytic continuation, functional identities, and reflection symmetry depend on the choice of branch. While the principal branch recovers the classical completed zeta function and its full suite of symmetries, all other branches exhibit structural asymmetries, isolated singularities, conjugation invariance, and a breakdown of functional symmetry. Despite these differences, the nontrivial zeros of the Riemann zeta function persist across all branches, forming a consistent zero set. Through rigorous analysis, we show that this consistency is preserved only when the zeros lie on the critical line, establishing a necessary condition for analytic coherence across the entire multibranch family.