Explicit formula and quasicrystal definition

Abstract

We show that the Riemann hypothesis is true if and only if the measure $$\hat{I}¼=-\sum _{n=1}^{\infty }\frac{\hat{I}›(n)}{\sqrt{n}}(\hat{I}{´}_{\log n}+\hat{I}{´}_{-\log n})+2\cosh (x/2)dx$$ is a tempered distribution. In this case it is the Fourier transform of another measure $$\mathcal{F}(\sum _{\hat{I}}^3\hat{I}{´}_{{\hat{I}}^3/2\ddot{I}€}-2{\vartheta }^{\prime }(2\ddot{I}€t)dt)=\hat{I}¼.$$ We propose a definition of Fourier quasi-crystal to make sense of Dyson suggestion.

6 pages. Corrected misprint in the proof that $\hat{I}¼$ tempered implies the RH, and give a fuller explanation of the proof