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when they exploit the relationship by Berry and Keating between the Riemann zeros and eigenvalues of random matrices

why do they choose $$ \frac{\gamma _{n}}{2\pi}log \frac{\gamma}{2\pi e} $$ as a Random variable

however why are they ignoring the number $ \frac{1}{\pi}arg \zeta (1/2+is) $

i am referring to the paper

http://www.maths.bris.ac.uk/~majpk/papers/67.pdf

why he just ignore the scillating term proportional to the argument of the zeta function of the critical lines.

Jose Garcia
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1 Answers1

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The normalization of zeros is chosen to make the number of zeta zeros with imaginary part in the interval $[0,W]$ behave like $W$ plus lower order terms, so that upon dividing by $W$ and taking the limit, you get a density function on the upper critical line. Why are you concerned about the argument of zeta?

The point is that if we now look at pair correlations of how far apart any given pair of zeros ought to be, then this behaves in a very similar way to the pair-correlations in random unitary matrices.

Alex R.
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  • but the term $ arg \zeta (1/2+it) $ is also important because you can have thousands of operators with real or imaginary eigenvalues so the mean density of zeros is about $ N(T)= \frac{T}{2\pi}log \frac{T}{2\pi e} $ this does not prove all the zeros are real. – Jose Garcia Sep 17 '13 at 09:14
  • I think it's mainly the scaling they are after. The point is they aren't looking for the Hilbert Polya operator which has exactly the zeros of the zeta function but they are interested in showing that the behavior is similar to the repulsion one sees in random matrix eigenvalues. Essentially, as one takes the limit to infinity the average behaviors are the same meaning that any specific behavior to the zeta function gets washed out. – Alex R. Sep 17 '13 at 15:48