The author has previously extended the theory of regular and irregular
primes to the setting of arbitrary totally real number fields. It has been
conjectured that the Bernoulli numbers, or alternatively the values of the
Riemann zeta function at odd negative integers, are evenly distributed
modulo p for every p. This is the basis of a well-known heuristic given by
Siegel for estimating the frequency of irregular primes. So far, analyses
have shown that if Q(\sqrt{D}) is a real quadratic field, then the values
of the zeta function \zeta_{D}(1-2m)=\zeta_{Q(\sqrt{D})}(1-2m) at negative
odd integers are also distributed as expected modulo p for any p. We use
this heuristic to predict the computational time required to find quadratic
analogues of irregular primes with a given order of magnitude. We also
discuss alternative ways of collecting large amounts of data to test the
heuristic.