  We discuss in some detail the general problem of computing averages of
convergent Euler products, and apply this to examples arising from singular
series for the $k$-tuple conjecture and more general problems of polynomial
representation of primes. We show that the singular series for the $k$-tuple
conjecture have a limiting distribution when taken over $k$-tuples with
(distinct) entries of growing size, and observe that its moments have a curious
symmetry property. We also give conditional arguments that would imply that the
number of twin primes (or more general polynomial prime patterns) in suitable
short intervals are asymptotically Poisson distributed.
