## Double Sum

A nested sum over two variables. Identities involving double sums include the following:

 (1)

where
 (2)

is the Floor Function, and
 (3)

Consider the sum

 (4)

over binary Quadratic Forms. If can be decomposed into a linear sum of products of Dirichlet L-Series, it is said to be solvable. The related sums
 (5) (6) (7)

can also be defined, which gives rise to such impressive Formulas as
 (8)

A complete table of the principal solutions of all solvable is given in Glasser and Zucker (1980, pp. 126-131).

Glasser, M. L. and Zucker, I. J. Lattice Sums in Theoretical Chemistry.'' Theoretical Chemistry: Advances and Perspectives, Vol. 5. New York: Academic Press, 1980.
Zucker, I. J. and Robertson, M. M. A Systematic Approach to the Evaluation of .'' J. Phys. A: Math. Gen. 9, 1215-1225, 1976.