Using
CLT, I predicted the distribution of LogZeta in the neighbourhood of any root
of Zeta is Rayleigh^{1}
(Section 6). Keating/Snaith have shown the real and imaginary parts
of log{ ζ(½ + iT)} are independent, identical Normal distributions as
height on the Critical Line T→∞ and this result has been known for
some time.

In
the neighbourhood of a zero (and anywhere on or off the Critical Line Re(z)=½), in the asymptotic
limit, the magnitude of LogZeta = R =√(X^{2} + Y^{2}) can
be shown Rayleigh distributed where X and Y are identical, mutually independent
Normal distributions each with mean zero N (0, 1); variance is one by
normalization. In Sonar, which is my area of expertise, they are said to be
‘LogNormal’ (in Zeta). (Interestingly, Real and Imaginary spectral components
have actually been measured in all manner of variety of physical systems that
involve thermal noise).

A treatment of the various statistical distributions in sonar and radar is provided by clicking on the following link, which gives a nice derivation of the Ricean distribution.

It is worth mentioning the treatment of spectrum components in Sonar. If X and Y are each the sum of real and imaginary spectral components respectively. Then X and Y are mutually independent in the limit as the number of samples n →∞, even if the individual components themselves have dependency on others. The reason for this is as follows:

Each
of the real and imaginary sums are composed of a series of trigonometric cosines
and sines respectively. Inverse trigonometric functions always have 2 values
between the range [0, 2p]
e.g. if cos(q1)
= 0, then q1
= 90° or 270° i.e. 2 possibilities for the sin(q1)
namely ±1. Given known values for each cos(q1)
and cos(q2)
then there are 4 possible values for {sin(q1)
+ sin(q2)}.
With an infinite series, the total possibilities for the sum of sine terms
become endless, so that the X and Y become increasingly uncorrelated and thus
independent in the limit

The
level spacings d_{i}=x _{i+j} -x_{j} are as actually
distributed (asymptotically as T→∞) as follows:

All material on this site is property of Adrian Rifat.