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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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References:
[1] Andrews G. E., Trans. Amer. Math. Soc. 293 pp 113– (1986)
[2] Andrews G. E., J. Number Theory 23 pp 285– (1986) · Zbl 0586.10007
[3] Bailey D. H., Experiment. Math. pp 17– (1994) · Zbl 0810.11076
[4] Borwein J. M., Pi and the AGM: A study in analytic number theory and computational complexity (1987) · Zbl 0611.10001
[5] Borwein D., J. Math. Anal. Appl. 188 pp 209– (1994) · Zbl 0814.40003
[6] Borwein J., Experimental Mathematics (2000)
[7] Borwein J. M., Electron. J. Combin. (1996)
[8] Borwein D., J. Math. Phys. 26 (11) pp 2999– (1985) · Zbl 0587.40007
[9] Borwein D., Trans. Amer. Math. Soc. 350 pp 3131– (1998) · Zbl 0932.11064
[10] Borwein J., ”Special values of multidimensional polylogarithms” (1998)
[11] Buhler J. P., J. Phys. A Math. Gen. 23 (12) pp 2523– (1990) · Zbl 0727.11024
[12] Buhler J., Mathematica in Education and Research pp 49– (1996)
[13] Crandall R. E., Topics in advanced scientific computation (1996) · Zbl 0844.65001
[14] Crandall R. E., Math. Comp. 67 (223) pp 1163– (1998) · Zbl 0901.11036
[15] Crandall R., ”Fast evaluation of Epstein zeta functions” (1998)
[16] Crandall R. E., J. Phys. A Math. Gen. 20 (16) pp 5497– (1987) · Zbl 0675.33009
[17] Crandall R. E., Experiment. Math. pp 275– (1994) · Zbl 0833.11045
[18] Crandall R. E., J. Phys. A Math. Gen. 20 pp 2279– (1987)
[19] Forrester P. J., J. Phys. A Math. Gen. 15 pp 911– (1982)
[20] Glasser M., Theor. Chem. Adv. Persp. pp 67– (1980)
[21] Kukhtin V. V., J. Phys. A Math. Gen. 26 pp L963– (1993)
[22] Zucker I. J., J. Phys. A Math. Gen. pp 499– (1976) · Zbl 0321.33016
[23] Zucker I. J., SIAM J. Math. Anal. 15 pp 406– (1984) · Zbl 0538.33001
[24] Zucker I. J., J. Phys. A Math. Gen. 20 pp L13– (1987) · Zbl 0638.33008
[25] Zucker I. J., J. Phys. A Math. Gen. 23 pp 117– (1990) · Zbl 0744.11048
[26] Zucker, I. J. 1998. [Zucker 1998], private communication
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