Reference Request: Convergence of Series over Integer Lattice I am writing a school paper on Modular forms, and I want to use the following result without including a proof, since it is not directly related to my main topic. 

For $s>0$, $d \in \mathbb{N}$, the series $\sum\limits_{v\in\mathbb{Z}^d\setminus\{0\}} \frac{1}{{\left \| v \right \|}^s}$ converges iff $s>d$.

However I am struggling to find a good reference for this to include in my paper. Can anyone point me to a book/paper that proves this?

It is enough to prove the convergence of the sum over tuples of nonzero integers. 
$$
\sum_{v\in (\mathbb{Z}-\{0\})^d} \frac1{\|v\|^s} \ \ \ \tag{1}
$$
Apply the AM-GM inequality, we have for $v=(v_1,v_2,\ldots, v_d)$, 
$$
\frac1{\|v\|^2} \leq \frac d{(v_1^2v_2^2\cdots v_d^2)^{1/d} }
$$
The $s/2$-th powers of these satisfy
$$
\frac1{\|v\|^s}\leq \frac{d^{s/2}}{|v_1v_2\cdots v_d|^{s/d}}
$$
If $s>d$, then the sum $(1)$ converges by comparison test.
