Convergence of Madelung-like lattice sums

DOI10.1090/S0002-9947-98-01983-7zbMath0932.11064OpenAlexW2142379291MaRDI QIDQ4211095

David Borwein, Jonathan M. Borwein, Christopher G. Pinner

Publication date: 10 September 1998

Published in: Transactions of the American Mathematical Society (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1090/s0002-9947-98-01983-7

zbMATH Keywords

theta functionsDirichlet serieszeta functionsconditional convergenceMadelung's constantLattice sums

Mathematics Subject Classification ID

Statistical mechanics of crystals (82D25) Convergence and divergence of series and sequences (40A05) Lattice points in specified regions (11P21) Zeta functions and (L)-functions (11S40) Dirichlet series, exponential series and other series in one complex variable (30B50) Multiple sequences and series (40B05)

Related Items (7)

Integral representations for elliptic functions ⋮ Periodic discrete energy for long-range potentials ⋮ Maximal theta functions universal optimality of the hexagonal lattice for Madelung-like lattice energies ⋮ Convergence analysis of the Wolf method for Coulombic interactions ⋮ New Representations for the Madelung Constant ⋮ Two-dimensional, phase modulated lattice sums with application to the Helmholtz Green’s function ⋮ The Electrostatic Potential of Periodic Crystals

Cites Work

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• Analysis of certain lattice sums
• Remarks on a lattice point problem
• On the convergence problem for lattice sums
• Convergence of lattice sums and Madelung’s constant
• Lattice points in large convex bodies, II
• Some properties of Dirichlet L-series
• A systematic approach to the evaluation of Σ_(m,n>0)(am²+bmn+cn²)^-s
• Exponential Sums and Lattice Points II
• On a method of $\Omega$-estimates

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