extrema of |zeta (1/2 +i*t)| : some statistics

The Riemann Hypothesis implies the Lindelöf hypothesis, which states
that for any constant c>0, we have, for positive t:

|zeta(1/2 + i*t)| = O(t^c) .

The Riemann-Siegel Z function , Z: R -> R, has the
property that
Z(t) = +/- |zeta(1/2+i*t)| for all real t.

Good approximations to Z(t) can used to find all zeros of zeta on
the critical line Re(z) = 1/2 with imaginary part in [0, T],
for a large real T. Techniques are known to count all zeros
of zeta in the part of the critical 0<=Re(z)<=1 and 0<=Im(z)<=T.
When the two results agree, RH is verified "up to height T".

-----

Recently I became interested in extrema of Z(t) with a large
magnitude relative to t. Here large may mean that
|Z(t)| is 1 to 5 times log(t) for t up to 10^8, depending on t.

My data for maxima of Z(t) can be found here:
< http://www.geocities.com/ezcos/datav3b.html >

On values of t with record-breaking Z values are recorded.
A line such as:
max: t=1087277.000000 Z=41.549929 sigmas=2.989378

means that Z(1087277) ~= 41.55, and there is a local maximum of
Z(t) "close" to 1087277.0 (cf. source code for more details).
My program gives the approximation:
Z(1087277)/ln(1087277) ~= 2.99 [ so-called sigmas ].

A graph of the data can be found here:
< http://www.geocities.com/ezcos/Zmax3.jpg >
From t= 10^5 to t=10^8, the largest record values are
quite close to 4.07 * t^(0.1696) . Of course, this says nothing
about large values of Z(t) for t=10^20 or 10^30 ...

The source code in C can be found here:
< http://www.geocities.com/ezcos/zmax_source_c.html > .
It's free for anybody to do anything with.

In the near future, I think I'll try the Monte Carlo method to
evaluate Z(t): just pick values of 'long i' < msafe at
random, and abort the summation if it seems Z(t) will
not be large relative to t. Also, cosine terms
with i just under msafe vary slowly with t:
the vtheta(t) almost offsets the i*log(t) ...

I'm also wondering if the distribution of the maxima of
Z(t) for t in [T, T+ T/1000] could have simple approximations,
found by experimentation.

David Bernier
.