New Relations on the Critical Line: Riemann Zeta Zeros, Divergent Series, and Infinite Numbers

In this work, a formal asymptotic framework based on infinite number expressions is employed to investigate structural relations associated with the Dirichlet representation of the Riemann zeta function. Within this framework, infinite number objects are interpreted through asymptotic representatives and serve as symbolic encodings of asymptotic behavior in the regime x → ∞. A divergent real series is constructed from the sum of entries of an n × n matrix in the asymptotic limit n → ∞ and analyzed in relation to the squared modulus of a Dirichlet-type series. When the common parameter coincides with the imaginary part of a nontrivial zero of the Riemann zeta function on the critical line, the framework yields a structured cancellation mechanism, leading to parameter-dependent decay or convergence toward the constant −γ/2. Additional formal asymptotic relations are derived linking nontrivial zeros, divergent expressions, and the Euler–Mascheroni constant. The theoretical analysis is accompanied by numerical computations in double-precision arithmetic, which serve as consistency checks of the predicted asymptotic behavior. The proposed approach provides a coherent representative asymptotic methodology for organizing and analyzing identities involving divergent expressions arising in analytic number theory. The resulting relations are interpreted within this representative framework and are intended as structural asymptotic identities rather than classical equalities of divergent series.