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I found a theorem about multigraded Hilbert series stated as follows:

Let $R$ be a Noetherian multigraded algebra $R:=\bigoplus_{j\in\mathbb{N}^m}{R_j}$ over $R_0=\mathbb{C}$. If $R$ is generated by $h$ homogeneous polynomials of multidegrees $\alpha_1,\dots,\alpha_h$, then its Hilbert series $HS_R(\mathbf{t})$ can be expressed as $$HS_R(\mathbf{t})=\frac{P(\mathbf{t})}{\prod_{\ell=1}^h{(1-\mathbf{t}^{\alpha_\ell})}},$$ where $P(\mathbf{t})$ is a polynomial with integer coefficients. Furthermore, if $R$ is freely generated, then $P(\mathbf{t})=1$. (here $\mathbb{t}^{\alpha}=t_1^{\alpha_1}\cdots t_n^{\alpha_n}$).

I can no longer find the place where I read this. If anyone could provide me with a reference, that would be greatly appreciated.

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    $\begingroup$ The proof of Theorem 11.1 in Atiyah and Macdonald, Introduction to Commutative Algebra, carries over to the multigraded case. It is also an easy consequence of the Hilbert syzygy theorem. $\endgroup$
    – Richard Stanley
    Commented May 13, 2015 at 23:56

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As @Richard points out, this follows easily from proof with m = 1. In particular, you can do an induction on h and use the fact that length is additive on short exact sequences. However if you are looking for a specific reference, this paper might be of interest, but note again as @Richard said, the proof is to just write down Hilbert's original argument and see that it carries over. I hope this helps.

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