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New representations for the Madelung constant. (English) Zbl 0949.11062
Madelung’s constant, essentially the binding energy density of an ideal sodium-chloride crystal, can be defined to be the value at $$1/2$$ of the analytic continuation of a three-dimensional Epstein zeta function, namely $$M(s)=\sum'{(-1)^{x+y+z}\over (x^2+y^2+z^2)^s}$$. It is an open problem whether the Madelung constant $$M(1/2)$$ can be evaluated in terms of other, more basic mathematical constants such as zeta or gamma function values. In this paper the author, after giving a short historical overview, uses a modern theta function identity due to G. Andrews to derive several new, “almost closed-form” representations for the Madelung constant and in fact for the function $$M(s)$$. The Andrews identity is used to split $$M(s)$$ into several parts, some of which can be evaluated explicitly, while others are interesting number-theoretic functions in their own right. This also leads to some new integral representations for $$M(1/2)$$.

##### MSC:
 11Y35 Analytic computations 11E45 Analytic theory (Epstein zeta functions; relations with automorphic forms and functions) 11E25 Sums of squares and representations by other particular quadratic forms
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