# Nearest neighbour spacing distribution in the GUE

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## Power series expansion of the d.e. about the origin

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```     3       5        6          7          8                  2   9          10                2   11
-s       s        s          s          s       (-350 + 3 Pi ) s      491 s        (1694 - Pi ) s
----- + ------ - ------- - -------- + -------- + ----------------- - ------------ + ----------------
12 Pi   360 Pi         2   20160 Pi          2                3                 2                3
432 Pi               5400 Pi       5443200 Pi       63504000 Pi     239500800 Pi
```

## Power series expansion of nn(s) about s=0

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```      2  2       4  4       6  6        6  7         6              5       7               8  9
2 Pi  s    4 Pi  s    2 Pi  s    32 Pi  s     -8 Pi          -4 Pi    2 Pi     8   2144 Pi  s
-------- - -------- + -------- - --------- + (------ - 2 Pi (------ + -----)) s  + -----------
3          45        315        2025        243            243     14175          496125
```

## Numerical solution of the d.e., calculation of the mean and comparison with power series expansion

Here the power series solution at s=1.1 is used as the
initial condition

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```  {{y -> InterpolatingFunction[{1.1, 16.}, <>]}}
```

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Here the numerical solution is integrated to
determine the mean of the distrubution

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```  0.725227
```

Next the numerical solution is compared to the power series solution

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```  -Graphics-
```

The function nn(s) can be tabulated

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## Comparison with data from the GUE

Code to generate GUE matrices and plot empirical value of nn(s) (output suppressed)

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Calculation of theoretical curve for nn(s) and comparison with numerical data

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```  -Graphics-
```

## Comparison with data from zeros of Riemann zeta function for 10^6 zeros about the 10^20 zero, 10^6 zero and first zero

Here we read in the data as a list. The data gives the number
of zeros in the intervals [j*.05, (j+1)*.05) (j=0,...40). The third,
fourth and fifth columns contain the data for zeros about
zero 1, zero 10^6 and zero 10^20 respectively. The theoretical
value of nn(s) is plotted from the Table t.

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```  -Graphics-
```