# multi-index Dirichlet series

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.

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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.

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It's also my impression that the field is developing. (i.e I haven't found any books on the subject) –  maki Nov 19 '09 at 6:00
Those are some great findings of you! Thank you very much for that. I also share your impression about this field being still fresh and developing. It is my firm belief (nothing substantial on paper yet) that it is possible to deduce important properties of analytic continuations of Dirichlet series (like singularities, order of magnnitude etc) from the convergence behavior of higher-dimensional Dirichlet series. –  efq Nov 20 '09 at 21:29
Or even more general, if you take n-dimensional Dirichlet series and its analytical continuation, you can find about singularities by observing the convergence behavior of at most (2n)-dimensional Dirichlet series. –  efq Nov 20 '09 at 21:29

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.

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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.

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If I recall correctly, Stanley doesn't talk about Dirichlet generating functions. He certainly doesn't in Volume 1 of Enumerative Combinatorics; it's possible he does in Volume 2, I suppose, but I don't own Volume 2 so I can't check. –  Michael Lugo Nov 16 '09 at 22:27
Wilf has introductory material in his Generatingfunctionology (2nd ed.), § 2.6, starting from a formal power series standpoint. I'll keep looking for old notes and papers, but I'm guessing all this stuff is nicely tucked away in Maple or Mathematica by now. –  Jon Awbrey Nov 17 '09 at 17:26
Actually, combinatorics is not really my field, but I happened to find both volumes of Stanley´s enumerative combinatorics, and from what I have seen, he doesn´t mention multiple Dirichlet series anywhere. I have already looked in Generatingfunctionology, but it seems to discuss only one-dimensional Dirichlet series (and the basics of them). –  efq Nov 20 '09 at 21:20
Yes, still looking. It's been a while, but my recollection is that adding more variables from the formal power series standpoint was no essential complication, so long as the algebraic side of the manipulations made sense. Are you running into a lot that depends on the analytic side of things? –  Jon Awbrey Nov 20 '09 at 22:26
Yes, the analytic side is of great importance here as the main idea is that it is possible to find out about singularities and order of meromorphic continuations of Dirichlet series by studying the convergence of suitable higher-dimensional Dirichlet-series. At least, this seems to be the case for simple and double Dirichlet-series. Thus, some generalization is thinkable. –  efq Nov 21 '09 at 19:57

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.

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