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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Convergence of Madelung-like lattice sums


Authors: David Borwein, Jonathan M. Borwein and Christopher Pinner
Journal: Trans. Amer. Math. Soc. 350 (1998), 3131-3167
MSC (1991): Primary 11P21, 40A05; Secondary :, 11S40, 40B05, 82D25, 30B50
MathSciNet review: 1433111
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Abstract: We make a general study of the convergence properties of lattice sums, involving potentials, of the form occurring in mathematical chemistry and physics. Many specific examples are studied in detail. The prototype is Madelung's constant for NaCl:

\begin{equation*}\sum _{-\infty}^{\infty} \frac{(-1)^{n+m+p}} {\sqrt{n^2+m^2+p^2}} = -1.74756459 \cdots, \end{equation*}

presuming that one appropriately interprets the summation proccess.


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Additional Information

David Borwein
Affiliation: Department of Mathematics, University of Western Ontario, London, Ontario N6A 5B7, Canada
Email: dborwein@uwo.ca

Jonathan M. Borwein
Affiliation: Centre for Experimental and Constructive Mathematics, Simon Fraser University, Burnaby, BC V5A 1S6, Canada
Email: jborwein@cecm.sfu.ca

Christopher Pinner
Affiliation: Centre for Experimental and Constructive Mathematics, Simon Fraser University, Burnaby, BC V5A 1S6, Canada & Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada
Email: pinner@cecm.sfu.ca

DOI: http://dx.doi.org/10.1090/S0002-9947-98-01983-7
PII: S 0002-9947(98)01983-7
Keywords: Lattice sums, zeta functions, conditional convergence, Madelung's constant, Dirichlet series, theta functions
Received by editor(s): August 21, 1995
Received by editor(s) in revised form: June 24, 1996
Additional Notes: The first and second authors were partially supported by the Natural Sciences and Engineering Research Council of Canada. The second author also received support from the Shrum Endowment at Simon Fraser University.
Article copyright: © Copyright 1998 American Mathematical Society