I came across quasicrystals in the Wikipedia page for the Riemann Hypothesis and then followed the references. On page 215 of Birds and Frogs Dyson makes the claim

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

  1. Which definition for quasi-crystal is being used for "according to definition"?

  2. 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!

  • $\begingroup$ Search on MSE it has been discussed already. It is just saying under the RH, the Riemann explicit formula shows the imaginary parts of the non-trivial zeros and $n^{-1/2}\Lambda(n)$ are almost a Fourier transform pair. $\endgroup$ – reuns Oct 27 '17 at 3:20
  • $\begingroup$ It becomes a little more obvious once you noticed $\zeta(s+a)$ is the Laplace transform of the distribution $\sum_{n=1}^\infty \delta(u-\ln n)n^{-a}$ $\endgroup$ – reuns Oct 27 '17 at 3:27
  • $\begingroup$ Again, I can't help you if you don't explain what you don't understand. $\endgroup$ – reuns Oct 27 '17 at 23:56

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