multi-index Dirichlet series - MathOverflow

multi-index Dirichlet series

2009-11-16T21:33:47Z

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.

Answer by Jon Awbrey for multi-index Dirichlet series

2009-11-16T22:22:35Z

I used to study enumerating generating functions, mostly for various families of graphs, that allowed a mix of ordinary and exponential variables for tracking different kinds of additive weights along with dirichlet variables for tracking multiplicative weights. I don't remember there being a lot of literature — this was a few years back — but there was some. I'm guessing you already looked in the bibs of Stanley or Goulden and Jackson and so on? Will see if I can dig up some notes, but probably easier just persisting in your web searches.

Answer by maki for multi-index Dirichlet series

2009-11-19T05:59:44Z

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.

Answer by mvg for multi-index Dirichlet series

2010-01-05T09:20:03Z

I don´t know about general multi-index Dirichlet series, but there is a good amount of theory on multiple zeta-functions (special cases of what you are asking for). There is plenty of stuff in MathSciNet on this.

Answer by Anweshi for multi-index Dirichlet series

2010-01-05T11:09:20Z

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.