TITLE: multi-index Dirichlet series
QUESTION [8 upvotes]: Hi,
I have recently got interested in multi-index (multi-dimensional) Dirichlet series, i.e. series of the form $F(s_1,...,s_k)=\sum_{(n_1,...,n_k)\in\mathbb{N}^k}\frac{a_{n_1,...,n_k}}{n_1^{s_1}...n_k^{s_k}}$. I found some papers suggesting that multi-index Dirichlet series are in fact a distinct subfield for itself within analytic number theory. So, I´m now looking for some 'basic' learning materials/books or similar on this subject.
Any suggestions are greatly appreciated!
efq
PS: I believe I have already checked most books on multi-dimensional complex analysis/several complex variables.

REPLY [2 votes]: See P. Deligne, Multizeta values, Notes d'exposes, IAS Princeton, for the deep mathematical aspects of this. 
Also for a general relevance philosophy, see Kontsevich and Zagier, Periods, Mathematics Unlimited(2001). An electronic version is available here.
There are various references, including those of Zudilin, Cartier, Zagier, Terasoma, Oesterle(On polylogarithms), Manin(iterated integrals and ....). Please look into mathscinet. 
There seem to be many papers by Dorian Goldfeld and collaborators, too.