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 [56 (2): 212–223, 2009]. Here it is from a Wikipedia entry:
"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."