I recently encountered multi-variable generalizations of various classical zeta functions. For example, the multi-variable Riemann zeta function
$$ \zeta(s_1, \ldots , s_r) := \sum_{0 < n_1 < \cdots < n_r} \frac{1}{n_1^{s_1} \cdots n_r^{s_r}}, $$
or, for parameters $a_1, \ldots, a_r \in [0,1)$, the multi-variable Hurwitz zeta function
$$ \zeta(s_1, \ldots, s_r; a_1, \ldots, a_r) = \sum_{0 < n_1 < \cdots < n_r} \frac{1}{(n_1 + a_1)^{s_1} \cdots (n_r + a_r)^{s_r}} $$
or for fixed complex $\alpha, \beta$ satisfying some conditions, the double variable Barnes zeta function
$$ \zeta(s_1, s_2; \alpha, \beta) = \sum_{m = 0}^\infty \frac{1}{(m + \alpha)^{s_1}} \sum_{n = 0}^\infty \frac{1}{(m + \alpha + n \beta)^{s_2}}. $$
For reference, I'm seeing multi-var zeta function in a few places, the multi-var Hurwitz zeta function in Mehta and Viswanadham's Analytic continuation of multiple Hurwitz zeta function, and the double variable Barnes zeta functions in Matsumoto's Asymptotic expansions of double zeta functions of Barnes, of Shintani, and Eisenstein series (indeed, Matsumoto's equation (1.20) has a full multi-variable version which generalizes the double variable one).
All of this raises two questions for me as a number theorist:
- What arithmetic consequences follow from the study of such multiple variable zeta functions? I'm seeing in the aforementioned and adjacent papers that the multiple variable zeta functions enjoy properties analogous with their single-variable counterparts (e.g. analytic continuation, Bernoulli-like expressions for integral values, bounded growth of moments). But what makes classical zeta functions truly interesting to me is the way in which those properties correspond to interesting arithmetic info (e.g. Prime Number Theorems, or the Herbrand-Ribet theorem). I find myself wondering if multiple variable zeta functions are studied for the sheer purpose of generalization.
- Are there useful/interesting/meaningful notions of multi-variable $L$-functions attached to your favourite number theoretic object? For example, some kind of multi-variable Dedekind zeta function attached to a number field. A cursory Google search found me nothing, and I've not encountered multi-variable $L$-functions before.