A Colombeau--Beurling criterion for the Riemann hypothesis

Abstract

This paper establishes an equivalence between the Riemann hypothesis and the association of a single moderate net in the Colombeau algebra G(0,1), constructed from damped Baez-Duarte sums. Two explicit damping strategies are introduced: an exponential damping exp(-k eps^2) combined with super-exponential truncation, and a polynomial damping k^(-delta(eps)), where delta(eps) = (log(1/eps))^(-alpha), combined with polynomial truncation. Assuming the Riemann hypothesis, the corresponding nets are shown to be moderate and associated with the negative characteristic function of (0,1). Conversely, the existence of a moderate net of this form associated with the negative characteristic function of (0,1) implies the Riemann hypothesis. The result provides a Colombeau-Beurling type criterion that reformulates the Riemann hypothesis in terms of generalized functions and weak association.