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With the p.d.f. for the infinite GUE now evaluated, we can further
test the GUE hypothesis by calculating for sequences of consecutive
zeros of the Riemann zeta function, using the data of .
Three sets of consecutive zeros
were analyzed, the data sets starting at zero number
, and respectively. The
quantity , where
with denoting the smoothed local density of zeros at
, was calculated and a histogram was constructed for the number of
values out of the that were tested that fell into the intervals
, . In Figure 2 the corresponding
empirical values of
at the points are plotted as a histogram
and compared with the value of
for the infinite GUE. The convergence towards the GUE value as the
magnitude of the imaginary part increases is evident.
For further comparison the moments
were calculated and compared with the empirical data according to the law
of large numbers prediction
The results are contained in Table 1. Again the trend is towards convergence to
the GUE value. Note in particular the four figure agreement between and .