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.



multi-index Dirichlet series
De la Breteche proved recently a Tauberian theorem for multiple Dirichlet series (MR1858338 (2002j:11106)). This is useful stuff in applications. It fails shortly of proving the main result in Balazard, et. al recent paper: <http://iml.univ-mrs.fr/~balazard/pdfdjvu/19.pdf> (but does so assuming the Riemann Hypothesis). Finally Daniel Bump (look up his homepage on google) did a lot of work on multiple Dirichlet series - unfortunately I am not familiar with any of it - it also seems to have a more algebraic flavor to it. 


P.S: It is remarkable that De La Breteche avoids using several complex variables.


