FLINT3.7.0-dev
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acb_mzeta_madelung_nacl

Synopsis#include <flint/acb_mzeta.h>
void acb_mzeta_madelung_nacl(arb_t res, slong prec)source

Sets res to the Madelung constant of sodium chloride,

M=−∑(i,j,k)≠0(−1)i+j+ki2+j2+k2=1.7475645946331821906…,M = -\sum_{(i,j,k) \ne 0} \frac{(-1)^{i+j+k}}{\sqrt{i^2+j^2+k^2}} = 1.7475645946331821906\ldots,

computed as −ZZ3,1(0;(12,12,12))-Z_{\mathbb{Z}^3,1}\bigl(0; (\tfrac12,\tfrac12,\tfrac12)\bigr) by acb_mzeta_epstein_gram_shell, which here reduces to Crandall's expansion [Cra1999, eq. (1.2)] as an alternating sum of r3(N)r_3(N) times complementary error functions and a power series in e−2π2/λe^{-2\pi^2/\lambda} with coefficients r3(8k+3)/(8k+3)r_3(8k+3)/(8k+3); see The Madelung constant. The computation is parallel with up to flint_get_num_threads() threads, and the result does not depend on the number of threads.

References#

  • [Cra1999] R. E. Crandall, "New representations for the Madelung constant", Experimental Mathematics 8 (1999), 367-379
Cite this page

The FLINT team, “acb_mzeta_madelung_nacl (acb_mzeta.h),” FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev, 2026. https://flintlib.org/doc/acb_mzeta/acb_mzeta_madelung_nacl.html (generated 2026-11-07).