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I ran across a double sum and was wondering if someone may be adept at evaluating it. I must admit that my double summation skills could be better, and I am always ready to learn more.

Show that:

$$\sum_{n=1}^{\infty}\sum_{k=1}^{\infty}\frac{\cos(\frac{\pi}{3}(n-k))}{nk(n+k)}=\frac{-1}{9}\zeta(3)+\frac{\pi\sqrt{3}}{27}\psi_{1}(1/3)-\frac{2\pi^{3}\sqrt{3}}{81}$$

I found this while playing around with:

$$\int_{0}^{1}\frac{\log(1-xe^{\frac{\pi i}{3}})\log(1-xe^{\frac{-\pi i}{3}})}{x}dx$$

I would enjoy seeing a clever evaluation of the sum.

I tried breaking it up as:

$$\sum_{n=1}^{\infty}\sum_{k=1}^{\infty}\frac{\cos(\frac{\pi k}{3})\cos(\frac{\pi n}{3})}{kn^{2}}$$ $$-\sum_{n=1}^{\infty}\sum_{k=1}^{\infty}\frac{\cos(\frac{\pi k}{3})\cos(\frac{\pi n}{3})}{n^{2}(k+n)}$$ $$+\sum_{n=1}^{\infty}\sum_{k=1}^{\infty}\frac{\sin(\frac{\pi k}{3})\sin(\frac{\pi n}{3})}{kn^{2}}$$ $$-\sum_{n=1}^{\infty}\sum_{k=1}^{\infty}\frac{\sin(\frac{\pi k}{3})\sin(\frac{\pi n}{3})}{n^{2}(k+n)}$$

The third from the top evaluates to $$\frac{\pi}{3}Cl_{2}(\frac{\pi}{3})=\frac{\pi}{3}\left(\frac{\sqrt{3}}{6}\psi_{1}(1/3)-\frac{\pi^{2}\sqrt{3}}{9}\right)$$

I think the top one evaluates to 0.

The other 2 are a little more challenging.

Would anyone enjoy lending a hand and showing a method to evaluate said double sum...or even the integral for that matter?.

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