Asymptotic Regularization of Vacuum Energy Fluctuations via the Generalized Voronoi Summation of Rankin-Selberg L-Function Coefficients and Associated Spectral Statistics

Mathematical Applications of Science Fiction We present a rigorous derivation for the regularization of divergent vacuum energy densitiesin quantum field theories defined over arithmetic manifolds. By mapping the spectral densityof the Laplacian operator to the zeros of Rankin-Selberg L-functions, we invoke a generalizedVoronoi summation formula for GL (n) automorphic forms. We establish that the high-energyultraviolet cutoff, typically introduced ad hoc in renormalization procedures, arises naturallyas a non-trivial zero of a spectral zeta function associated with the automorphic cuspidalrepresentation. We prove five major theorems regarding the convergence of the regularizedstress-energy tensor, utilizing advanced techniques from the spectral theory of automorphicforms, including the Kuznetsov trace formula and the analytic continuation of Eisenstein series.The resulting framework provides a finite, unambiguous value for the vacuum expectationvalue (VEV) without recourse to subtraction schemes, suggesting a deep number-theoreticstructure underlying the fabric of spacetime vacuum fluctuations.