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    Lattice Sum

    Lattice sum (common versions)

    sum_(i=-∞)^∞ sum_(j=-∞)^∞ sum_(k=-∞)^∞ sum_(l=-∞)^∞ sum_(m=-∞)^∞ sum_(n=-∞)^∞ sum_(p=-∞)^∞ sum_(q=-∞)^∞( piecewise | (-1)^(i + j + k + l + m + n + p + q) (i^2 + j^2 + k^2 + l^2 + m^2 + n^2 + p^2 + q^2)^(-s) | i^2 + j^2 + k^2 + l^2 + m^2 + n^2 + p^2 + q^2!=0 0 | otherwise) = -16 (1 - 2^(4 - s)) ζ(s - 3) ζ(s) for Re(s)>1 sum_(i=-∞)^∞ sum_(j=-∞)^∞ sum_(k=-∞)^∞ sum_(l=-∞)^∞( piecewise | (-1)^(i + j + k + l) (i^2 + j^2 + k^2 + l^2)^(-s) | i^2 + j^2 + k^2 + l^2!=0 0 | otherwise) = -8 (1 - 2^(1 - s)) (1 - 2^(2 - s)) ζ(s) ζ(s - 1) for Re(s)>1 sum_(k=-∞)^∞ sum_(j=-∞)^∞( piecewise | (-1)^(j + k) (j^2 + k^2)^(-s) | j^2 + k^2!=0 0 | otherwise) = (1 - 2^(1 - s)) (-1) 2^(2 - s) ζ(s) Φ(-1, s, 1/2) for Re(s)>0 sum_(i=-∞)^∞ sum_(j=-∞)^∞ sum_(k=-∞)^∞ sum_(l=-∞)^∞ sum_(m=-∞)^∞ sum_(n=-∞)^∞( piecewise | (-1)^(i + j + k + l + m + n) (i^2 + j^2 + k^2 + l^2 + m^2 + n^2)^(-s) | i^2 + j^2 + k^2 + l^2 + m^2 + n^2!=0 0 | otherwise) = 2^(4 - s) (1 - 2^(1 - s)) Φ(-1, s - 2, 1/2) ζ(s) - 2^(4 - s) (1 - 2^(3 - s)) Φ(-1, s, 1/2) ζ(s - 2) for Re(s)>1

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