  In a one-page fragment published with his lost notebook, Ramanujan stated two
double series identities associated, respectively, with the famous Gauss Circle
and Dirichlet Divisor problems. The identities contain an "extra" parameter,
and it is possible that Ramanujan derived these identities with the intent of
attacking these famous problems. Similar famous unsolved problems are connected
with $f_K(n)$, the number of integral ideals of norm $n$ in an algebraic number
field $K$. In this paper we establish Riesz sum identities containing an
"extra" parameter and involving $f_K(n)$, or divisor functions associated with
$K$. Upper bounds for the sums as the upper index tends to infinity are also
established.
