Explicit formula and quasicrystal definition

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.