In some previous publications, Zucker and Robertson [13, 14, 15] for certain cases evaluated exactly the double sum
with a, b, c integers. In (1·1) the summation is over all integer values of m and n, both positive and negative, but excluding the case where m and n are simultaneously zero. The term ‘exact’ used here is in the sense introduced by Glasser , and means that S may be expressed as a linear sum of products of pairs of Dirichlet L-series. S is then said to be solvable. Whether S may be solved or not depends on the properties of the related binary quadratic form (am2 + bmn + cn2) = (a, b, c). The cases considered in  were when a > 0 and d = b2 − 4ac < 0, i.e. (a, b, c) was positive definite. When this is so, Glasser  conjectured that S was solvable if and only if (a, b, c) had one reduced form per genus, i.e. the reduced forms of (a, b, c) were disjoint. Zucker and Robertson  were in fact able to solve S for all (a, b, c) for which d < 0 and whose reduced forms were disjoint. A complete list of solutions may be found in .