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Convergence of Madelung-like lattice sums. (English) Zbl 0932.11064

The authors investigate convergence properties of sums of the form \[ A_\nu(s):= \sum_{(x_1,\dots, x_k)\in C_\nu} \frac{(-1)^{x_1+\cdots+ x_k}} {Q(x_1,\dots, x_k)^s} \] where \(Q\) is a positive definite symmetric quadratic form and \(C_\nu= \nu C\cap (\mathbb{Z}^k \setminus 0)\) with \(C\subseteq \mathbb{R}^k\) bounded. The prototype is Madelung’s constant for NaCl: \[ \sum_{-\infty}^\infty \frac{(-1)^{n+m+p}} {\sqrt{n^2+m^2+p^2}}= -1.74756459\dots, \] which depends on the summation process. The main result is that \(\lim_{\nu\to\infty} A_\nu(s)\) exists and is analytic down to \(\text{Re } s>(k-1)/2\) for all reasonably shaped regions \(C\) (e.g. convex in one of the variables), and hence that the limit is independent of such \(C\). Moreover, they prove that for any specific regions \(C\) (e.g., certain ellipses or polygons) the area of convergence can be extended; for rectangles even down to \(\text{Re } s>0\).

MSC:

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