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arXiv · 2402.10604

Explicit formula and quasicrystal definition

J. Arias de Reyna
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Abstract

We show that the Riemann hypothesis is true if and only if the measure $$\mu=-\sum_{n=1}^\infty\frac{\Lambda(n)}{\sqrt{n}}(\delta_{\log n}+\delta_{-\log n})+2\cosh(x/2)\,dx$$ is a tempered distribution. In this case it is the Fourier transform of another measure $$\mathcal{F}\Bigl(\sum_{\gamma}\delta_{\gamma/2\pi}-2\vartheta'(2\pi t)\,dt\Bigr)=\mu.$$ We propose a definition of Fourier quasi-crystal to make sense of Dyson suggestion.

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J. Arias de Reyna. 2024-02-16. Explicit formula and quasicrystal definition. https://arxiv.org/abs/2402.10604

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