New Representations for the Madelung Constant

1. Overview of the Madelung Problem 2. Analytic Consequences of the Andrews Identity 3. Connection with Multiple Zeta Sums 4. An -Series for M(s) 5. Integral Representations for the Madelung Constant 6. Number-Theoretical Implications: Sums of Three Squares 7. Open Problems Acknowledgments References From a modern theta-function identity of G. E. Andrews we derive new representations for the celebrated Madelung constant and various of its analytic relatives. The method leads to connections with the modern theory of multiple zeta sums, generates an apparently entire “ series” representation, and, for the Madelung constant in particular, yields a finite-integral representation. These analyses suggest variants of the Andrews identity, leading in turn to number-theoretical results concerning sums of three squares.