In an article I am reading they obtain the following $$ \sum_{\mathbf{x}\in \mathbb{Z}^{2} \backslash \{0\}, \| \mathbf{x} \| \leq Z } \frac{1}{\sqrt{x_1^4 + x_1^2x_2^2 + x_2^4 } } = c Z(1 + O(1/Z)), $$ where $\| \cdot \|$ is the Euclidean norm and $$ c = \int_{\| \mathbf{t}\| \leq 1 } \frac{d \mathbf{t}}{\sqrt{t_1^4 + t_1^2t_2^2 + t_2^4 } }. $$ The only explanation is that it follows by an application of partial summation and elementary lattice point counting argument.
I am not sure how this holds and explanation would be appreciated. Thank you.