If the Riemann hypothesis is true, then the zeros of the zeta-function form a one-dimensional quasi-crystal according to the definition. They constitute a distribution of point masses on a straight line, and their Fourier transform is likewise a distribution of point masses, one at each of the logarithms of ordinary prime numbers and prime-power numbers.
Could someone please explain this statement? I am specifically confused on
Which definition for quasi-crystal is being used for "according to definition"?
Why is the Fourier Transform of the zeros of the zeta function also a distribution of the point masses at the logarithms of ordinary prime numbers and prime powers?
I have a decent knowledge of number theory, some abstract algebra, analysis, plenty of calculus, and have used Fourier Transforms in computer programs. Hopefully this is enough to understand the abstraction. Any help would be appreciated!