,
which arises in a quite natural way from the techniques
of Dirichlet polynomials. The other is the factor 
which arises from Random Matrix theory.
In the conjectures for mean-values there are two factors
to notice: one is the arithmetical function ,
which arises in a quite natural way from the techniques
of Dirichlet polynomials. The other is the factor
which arises from Random Matrix theory.
These conjectures are a kind of pasting together of random matrix theory techniques and Dirichlet polynomial techniques.
An explanation for this phenomenon of disparate pieces
is that the random matrix theory explains behavior of
zeros clustered together in a spacing of approximately
the inverse of the logarithm of the conductor, whereas it is
quite likely that the arithmetical arises from
considerations of longer range correlations of the zeros
of the family. The fact that the do not appear from
random matrix theory pose no problem from the point of view
of the highest order main term of the asymptotic formula.
However, we know that for the second and fourth moments
of 
the main terms are actually polynomials of
degrees 1 and 4, all terms of which can be expressed in
closed form (See "Lower Order terms") .
These terms seem to involve derivatives of at 
and 2 as well as mysterious geometric factors that should
be explainable from random matrix theory.
In order to really understand the mechanism connecting prime numbers, random matrices, and mean-value formulas, we will likely need to proceed via the explicit formula of Weil [ MR 14,727e] (see also Guinand [ MR 10,104g]). This formula is the explicit link between zero sums and prime number sums which will be needed to rigorously explain the connection between short and long range correlations.
Thus, we want a model of the Riemann zeta-function as a product of two factors: one of which is a kind of Euler product and the other is a kind of Hadamard product (whose behavior is like that of the characteristic polynomial of a random unitary matrix).
A related problem is to find a model for the distribution of zeros which works for both short and long range correlations. This would presumably include a random matrix model for the short range behavior, and some other model (somehow based on the prime numbers?) for the long range bahavior.
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