A Colombeau--Beurling criterion for the Riemann hypothesis

Summary · qwen2.5:32b

The paper establishes an equivalence between the Riemann hypothesis and the properties of a specific moderate net within the Colombeau algebra $G(0,1)$, constructed through damped Bézout sums; this reformulates the Riemann hypothesis using generalized functions and weak association criteria. Two explicit damping strategies are introduced: one with exponential damping $\exp(-k\varepsilon^2)$ and super-exponential truncation, and another with polynomial damping $k^{-\delta(\varepsilon)}$ where $\delta(\varepsilon)=(\log(1/\varepsilon))^{-\alpha}$, illustrating the conditions under which these nets are moderate and uniformly $L^2$-bounded.

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Mathematical paper investigates the Riemann hypothesis using Colombeau algebra and generalized functions.

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arXiv:2606.22562v2 Announce Type: replace 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.