Explicit formula and quasicrystal definition
J. Arias de Reyna
TL;DR
This work makes Dyson's quasi-crystal intuition precise by linking the zeros of the Riemann zeta function to a tempered-distribution Fourier pair. It proves that the Riemann hypothesis is equivalent to the temperedness of the measure $\mu = -\sum_{n=1}^\infty\frac{\Lambda(n)}{\sqrt{n}}(\delta_{\log n}+\delta_{-\log n})+2\cosh(x/2)\,dx$, which is the Fourier transform of $\nu = \sum_\gamma\delta_{\gamma/2\pi}-2\vartheta'(2\pi t)\,dt$; and conversely, temperedness of $\mu$ implies RH. The paper provides Delsarte-type explicit formulae to connect sums over zeros with primes via the von Mangoldt function and shows that $\mu$ is the second derivative of a slowly growing continuous function, situating Dyson's proposal within rigorous distribution theory. It also offers a concrete, distributional definition of a Fourier quasi-crystal and clarifies how the expected density corrections (via $2\cosh(x/2)$ and $\vartheta'(2\pi t)$) realize a quasi-crystal structure in this setting.
Abstract
We show that the Riemann hypothesis is true if and only if the measure $$μ=-\sum_{n=1}^\infty\frac{Λ(n)}{\sqrt{n}}(δ_{\log n}+δ_{-\log n})+2\cosh(x/2)\,dx$$ is a tempered distribution. In this case it is the Fourier transform of another measure $$\mathcal{F}\Bigl(\sum_γδ_{γ/2π}-2\vartheta'(2πt)\,dt\Bigr)=μ.$$ We propose a definition of Fourier quasi-crystal to make sense of Dyson suggestion.