I would like to consider the following sum:
$$\displaystyle \sum_{\substack{\ \ k_1, k_2,...,k_r \in \mathbb{Z}^d \backslash \{0\}} \\ {\ \ \ \ \ k_1 + k_2 + ... + k_r = 0}} \frac{1}{\|k_1\| \|k_2\| \cdot \cdot \cdot \|k_r\|},$$
where each $k_i$ is a non-zero vector in $\mathbb{Z}^d,$ $r \geqslant 2,$ and $\| \cdot \|$ denotes the usual Euclidean norm on $\mathbb{R}^d.$ I would like to know under which conditions this sum converges (dependent on $r$ and the dimension $d$).
If $r = 2,$ then this sum converges only if $d = 1,$ by comparison with the Riemann zeta function with normed argument (see the link below). But what about for other values of $r$? For instance, does the series
$$\displaystyle \sum_{\substack{\ \ k_1, k_2,k_3 \in \mathbb{Z}^d \backslash \{0\}} \\ {\ \ \ \ \ k_1 + k_2 + k_3 = 0}} \frac{1}{\|k_1\| \|k_2\| \|k_3\|}$$
converge in any dimension $d$?
I suspect that this result and this result might possibly be useful here, but I'm not sure how one might implement those results in justifying the convergence (or divergence) of this sum for general $r.$
In the case where $r=3,$ the condition $k_1 + k_2 + k_3 = 0$ lets us replace at most one of $k_i$ in the sum, but from that point, it doesn't seem like there is anything obvious you can do (mainly due to the $\|k_1 + k_2\|$ term appearing in the denominator, if one chooses to replace $k_3$). You can obtain other expressions involving terms like $\|k_1\|^2 \|k_2\|$ by using the triangle inequality, but I don't know what to do with the resulting sums.
Another idea I had was to split the sum into regions in which $\|k_i\| \geqslant \|k_j\|$ for some $i,j,$ and then try to compare it to the Riemann zeta function with normed argument (which I've linked to above). Alternatively, taking $r=4$ (or any even number), then it is possible to bound the sum by a pair of identical integrals over $\mathbb{R}^d,$ which may or may not converge.
Mathematica struggles to deal with the sum even if the indices range over small intervals. However, if $r=3$ and $d=1$, then we get a sum which may converge (it appears to increase very slowly). It appears to approach some value just above 14 in this case.