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I am interested in evaluating the following sum

$I\left(s\right)=\sum_{\left\{ m,n\right\} \neq\left\{ 0,0\right\} }\frac{1}{\left(l^{2}+lm+m^{2}\right)^{s}}.$

This is a Madelung Constants type sum arises in triangular/hexagonal lattice in two dimension.

(For a reference: The corresponding expression for a square lattice, similar sum is given by

$\sum_{\left\{ m,n\right\} \neq\left\{ 0,0\right\} }\frac{1}{\left(l^{2}+m^{2}\right)^{2}}$

This has a closed form answer/expression given in

https://math.stackexchange.com/a/198288/16078)

Boa_Constrictor
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1 Answers1

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I found the answer in this paper; https://aip-scitation-org.proxy.lib.ohio-state.edu/doi/pdf/10.1063/1.1666618 (Equation 4).

$\sum_{\left\{ m,n\right\} \neq\left\{ 0,0\right\} }\frac{1}{\left(l^{2}+lm+m^{2}\right)^{s}}=6\zeta\left(s\right)g\left(s\right),$

where

$g\left(s\right)=\sum_{n=0}\left(3n+1\right)^{-s}-\left(3n+2\right)^{-s}.$

Boa_Constrictor
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