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Level Spacings Distribution

The above curve is the Normalised Maxwell/Boltzmann or Rayleigh^2 distribution; [pdf α 4x2exp(-2x2)], with appropriate normalisation in x (~ ˝ √(π/√ 2)x1.03). My explanation predicts this using CLT. The thermal distribution is also known as the Maxwell/Boltzmann distribution, which has been extensively researched, by the likes of Jeans in the 1900’s.

This is an EXACT match to the Odlyzko curve of level spacings distribution (The Bakerian lecture, 1987) of the zeros of Zeta on the Critical Line. I would ask the reader to check this by eyeball against the original graph, with the appropriate normalisation, and ensuring that this does in fact match at the higher level. (Using the appropriate normalisation is analogous to converting linear units e.g. feet to metres but DOES NOT CHANGE THE SHAPE of the curve). Thus

MAXWELL BOLTZMANN DISTRIBUTION = COMPLEX HERMITIAN STATISTICS = LEVEL SPACINGS OF NON-TRIVIAL ZEROS OF ZETA

(I have my own explanation, which concerns only possible correlations between consecutive zeros in half cycles at the bit level, which I will not discuss here).

This is an interesting result2


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