This is well below the technical level of the replies you seek and deserve (and which others more knowledgeable will no doubt supply), but I can't resist mentioning Freeman Dyson's idea, which I encountered in his 
"Birds and frogs" article in the 
*Notices of the American Mathematical Society* \[<b>56</b> (2): 212–223, 2009].
Here it is from a [Wikipedia entry][1]:

>"The Riemann hypothesis implies that the zeros of the zeta function form a quasicrystal, meaning a distribution with discrete support whose Fourier transform also has discrete support. Dyson suggested trying to prove the Riemann hypothesis by classifying, or at least studying, 1-dimensional quasicrystals."


  [1]: http://en.wikipedia.org/wiki/Riemann_hypothesis