A Colombeau--Beurling criterion for the Riemann hypothesis

Abstract

This paper establishes an equivalence between the Riemann hypothesis and the association, together with uniform $L^2$-boundedness, of a single moderate net in the Colombeau algebra $G(0,1)$, constructed from damped B\'aez--Duarte sums by multiplicative (Mellin) convolution. Two explicit damping strategies are introduced: an exponential damping $\exp(-k\varepsilon^2)$ combined with super-exponential truncation, and a polynomial damping $k^{-\delta(\varepsilon)}$, where $\delta(\varepsilon)=(\log(1/\varepsilon))^{-\alpha}$, combined with polynomial truncation. Assuming the Riemann hypothesis, the corresponding nets are shown to be moderate, uniformly $L^2$-bounded, and associated with the negative characteristic function of $(0,1)$. Conversely, the existence of a moderate net of this form that is uniformly $L^2$-bounded and 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, weak association, and uniform $L^2$ control.