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The Riemann zeta function, the primon gas, and supersymmetry.

rantonels.github.io · 1,838 words · saved by 1 readers

The following is probably one of the weirdest unexpected bridges between abstract mathematics and theoretical physics I know of; the two weirdest features being that it's pretty simple to explain, and that the area of math it connects to is number theory. While the latter thing can occur occasionally, especially in the area of string theory, the former is basically close to impossible. A while back I was in a very ugly situation I needed to distract myself out of. I was struck by the fact that the Riemann zeta function looks a lot like a partition function from statistical mechanics. The Riemann zeta is: ζ(β)= ∑ n=1 ∞ n −β 𝜁 ( 𝛽 ) = ∑ 𝑛 = 1 ∞ 𝑛 − 𝛽 for Re(β)>1 𝑅 𝑒 ( 𝛽 ) > 1 of course, and then one extends analytically. The partition function of a system with energy levels E n 𝐸 𝑛 is given by Z(β)= ∑ n e −β E n 𝑍 ( 𝛽 ) = ∑ 𝑛 𝑒 − 𝛽 𝐸 𝑛 where β= 1 k B T 𝛽 = 1 𝑘 𝐵 𝑇 . Note that if you have a system with energy levels E n =logn 𝐸 𝑛 = log ⁡ 𝑛 , its partition fun

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