Experimental Mathematics

New representations for the Madelung constant

Richard E. Crandall


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 "$\eta$ 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.

Article information

Experiment. Math., Volume 8, Issue 4 (1999), 367-379.

First available in Project Euclid: 9 March 2003

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11Y60: Evaluation of constants
Secondary: 11E45: Analytic theory (Epstein zeta functions; relations with automorphic 11M35: Hurwitz and Lerch zeta functions 33E20: Other functions defined by series and integrals 33F05: Numerical approximation and evaluation [See also 65D20]


Crandall, Richard E. New representations for the Madelung constant. Experiment. Math. 8 (1999), no. 4, 367--379. https://projecteuclid.org/euclid.em/1047262358

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