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i have been reading or at least trying to understand how Connes get the density (approximate) of states

$ N(E)= \frac{E}{2\pi}log \frac{E}{2\pi}- \frac{E}{2\pi}+ \frac{7}{8}+ \frac{1}{\pi}arg \zeta(1/2+iE)$

from the Hamiltonian operator $ H=xp$

the 'smooth' part i know how it is evaluated , simply by computing $ \frac{1}{2\pi}\int dx \int dp H(E-xp) $ and using a certina Maslov index

However , how did he manage to get the oscillating part of the zeros ??? i mean $ \frac{1}{\pi}Arg \zeta (1/2+iE) $

i have been reading the approach

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You might want to edit your title. – user5117 Apr 26 '12 at 21:58
See Titchmarsh "Theory of the Riemann Zeta Function" (or any other book on the zeta function.) – Stopple Apr 26 '12 at 22:16

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