  Gross and Smith have put forward generalizations of Hardy - Littlewood twin
prime conjectures for algebraic number fields. We estimate the behavior of sums
of a singular series that arises in these conjectures, up to lower order terms.
More exactly, we find asymptotic formulas for smoothed sums of the singular
series minus one. Based upon Gross and Smith's conjectures, we use our result
to suggest that for large enough 'short intervals' in an algebraic number field
K, the variance of counts of prime elements in a random short interval deviates
from a Cramer model prediction by a universal factor, independent of K. The
conjecture over number fields generalizes a classical conjecture of Goldston
and Montgomery over the integers. Numerical data is provided supporting the
conjecture.
