Explicit formula and quasicrystal definition

We show that the Riemann hypothesis is true if and only if the measure $$\mu =-\sum _{n=1}^{\mathrm{\infty }}\frac{\mathrm{\Lambda }(n)}{\sqrt{n}}({\delta }_{\log n}+{\delta }_{-\log n})+2\cosh (x\mathrm{/}2)dx$$ is a tempered distribution. In this case it is the Fourier transform of another measure $$\mathcal{F}(\sum _{\gamma }{\delta }_{\gamma \mathrm{/}2\pi }-2{\vartheta }^{\mathrm{\prime }}(2\pi t)dt)=\mu \mathrm{.}$$ We propose a definition of Fourier quasi-crystal to make sense of Dyson suggestion.