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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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      Abstract

      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
      Additional Information: pubdom FALSE
      Uncontrolled Keywords: lattice sums, zeta functions, conditional convergence, Madelung's constant, Dirichlet series, theta functions
      Subjects: 11-xx Number theory > 11Sxx Algebraic number theory: local and $p$-adic fields
      30-xx Functions of a complex variable > 30Bxx Series expansion
      40-xx Sequences, series, summability > 40Axx Convergence and divergence of infinite limiting processes
      40-xx Sequences, series, summability > 40Bxx Multiple sequences and series
      11-xx Number theory > 11Pxx Additive number theory, partitions
      Faculty: UNSPECIFIED
      Depositing User: Users 1 not found.
      Date Deposited: 17 Nov 2003
      Last Modified: 13 Jan 2015 14:05
      URI: https://docserver.carma.newcastle.edu.au/id/eprint/104

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