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I know how to prove that for one variable $x\in\mathbb{R}$ : $$\sum_{n\in\mathbb{Z}/\left\{ 0\right\} }\frac{\cos\left(nx\right)}{n^{2}}=\frac{x^{2}}{2}-\pi x+\frac{\pi^{2}}{3}$$ as seen at: Series $\sum_{n=1}^{\infty}\frac{\cos(nx)}{n^2}$.

But what about the multivariable case? If $x\in\mathbb{R}^d$, then is there a neat expression for: $$\sum_{\vec{n}\in\mathbb{Z}^{d}/\left\{ 0\right\} }\frac{\cos\left(\vec{n}\cdot\vec{x}\right)}{\vec{n}^{2}}$$

Sebastiano
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Ori Shem-Ur
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    If $x=0$ this becomes $$\sum_{(m_1,\dots,m_d)\in \mathbb Z^d\setminus 0}\frac1{m_1^2+\cdots +m_d^2}$$ Then we can use the four square theorem to easily deduces that the sum doesn't converge when $n\geq 4.$ You might be able to prove it doesn't converge for $d=2,3.$ $x\neq 0$ will be harder, though. – Thomas Andrews Dec 03 '22 at 21:09

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