# Convergence of Madelung-Like Lattice Sums

Borwein, David and Borwein, Jonathan M. and Pinner, Christopher (1998) Convergence of Madelung-Like Lattice Sums. Trans. Amer. Math. Soc., 350 . pp. 3131-3167.

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We make a general study of the convergence properties of lattice sums, involving potentials, of the form occuring in Mathematical Chemistry and Physics. Many specific examples are studied in detail. The prototype is Madelung's constant for NaCl: $$\sum_{-\infty}^{\infty} \frac{(-1)^{n+m+p}} {\sqrt{n^2+m^2+p^2}} = -1.74756459 \cdots$$ presuming that one appropriately interprets the summation proccess.
Item Type: Article pubdom FALSE lattice sums, zeta functions, conditional convergence, Madelung's constant, Dirichlet series, theta functions 11-xx Number theory > 11Sxx Algebraic number theory: local and $p$-adic fields30-xx Functions of a complex variable > 30Bxx Series expansion40-xx Sequences, series, summability > 40Axx Convergence and divergence of infinite limiting processes40-xx Sequences, series, summability > 40Bxx Multiple sequences and series11-xx Number theory > 11Pxx Additive number theory, partitions UNSPECIFIED Users 1 not found. 17 Nov 2003 13 Jan 2015 14:05 https://docserver.carma.newcastle.edu.au/id/eprint/104