Multiplicative Colombeau algebras and the Nyman--Beurling criterion for the Riemann hypothesis
This paper establishes an equivalence between the Riemann hypothesis and the association, together with uniform $L^2$-boundedness, of a moderate net in a Colombeau-type algebra built from polynomially damped Báez-Duarte sums. The regularization is performed by multiplicative (Mellin) convolution, which respects the dilation symmetry of the Beurling functions and guarantees that every approximant lies in the $L^2$-closure of the Beurling space. The equivalence is unconditional under the Riemann hypothesis: it uses only the qualitative convergence of Báez-Duarte, Mazur's theorem, and the classical Nyman--Beurling criterion. As a separate quantitative refinement, we prove that under two additional hypotheses on the non-trivial zeros of $ζ$ (simplicity and separation), the damping error admits a power-law bound with an explicit constant. This refinement is independent of the equivalence and is not used in its proof. The exponential damping $e^{-k\eps^2}$ is discussed as an open problem.