LECTURE 1. FINITE MODELS 341{Jn I((~+ iJn) = 0, Jn> 0} ·
For the normalized level spacings, replace Jn by Jn = 2 ~ Jnlo9Jn> since we want
the mean spacing to be one. Here one needs to know that the number of Jn such
that Jn:::; Tis asymptotic to 2 ~TlogT as T---> oo.
Historically the connections between the statistics of the Riemann zeta zeros Jn
and the statistics of the energy levels of quantum systems were made in a dialogue of
Dyson and H. Montgomery over tea at the Institute for Advanced Study, Princeton.
Odlyzko's experimental results show that the level spacings iJ n - J n+i I, for large
n, look like that of the Gaussian unitary ensemble (GUE). See Figure 5.oq 0<D d~ c:
.. 0
dN c;;0 .0 0.5 1.^0 1.5 2.0 2.5 3.^0
normalized spacingF igure 5. From Cipra [22]. Odlyzko's comparison of level spacing of zeros
of the Riemann zeta function and that for GUE (Gaussian unitary ensemble).
See Odlyzko and Forrester [30]. The fit is good for the 1,041,000 zeros near
the 2 x 1020 zero.Note. The level spacing distribution for the eigenvalues of Gaussian unitary
matrices is not a standard function in Matlab, Maple or Mathematica. Sarnak [67]
and Katz and Sarnak [45] proceed as follows. Let K s : L^2 [0, 1] ---> L^2 [0, 1] be the
integral operator with kernel defined by
sin ( ns(~-y))
h s (x ' y) -- n(x-y) ' for s ~ 0.
--2-Approximations to this kernel have been investigated in connection with the un-
certainty principle (see Terras [8 3 ], Vol. I , p. 51). The eigenfunctions are spher-
oidal wave functions. Let E(s) be the Fredholm determinant det(I - Ks) and let