multi-index Dirichlet series - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T15:51:58Z http://mathoverflow.net/feeds/question/5739 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/5739/multi-index-dirichlet-series multi-index Dirichlet series ex falso quodlibet 2009-11-16T21:33:47Z 2010-02-01T19:33:46Z <p>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.</p> <p>Any suggestions are greatly appreciated!</p> <p>efq</p> <p>PS: I believe I have already checked most books on multi-dimensional complex analysis/several complex variables.</p> http://mathoverflow.net/questions/5739/multi-index-dirichlet-series/5744#5744 Answer by Jon Awbrey for multi-index Dirichlet series Jon Awbrey 2009-11-16T22:22:35Z 2009-11-16T22:22:35Z <p>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 &mdash; this was a few years back &mdash; 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.</p> http://mathoverflow.net/questions/5739/multi-index-dirichlet-series/6065#6065 Answer by maki for multi-index Dirichlet series maki 2009-11-19T05:59:44Z 2009-11-19T05:59:44Z <p>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: <a href="http://iml.univ-mrs.fr/~balazard/pdfdjvu/19.pdf" rel="nofollow">http://iml.univ-mrs.fr/~balazard/pdfdjvu/19.pdf</a> (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> <p>P.S: It is remarkable that De La Breteche avoids using several complex variables.</p> http://mathoverflow.net/questions/5739/multi-index-dirichlet-series/10796#10796 Answer by mvg for multi-index Dirichlet series mvg 2010-01-05T09:20:03Z 2010-01-05T09:20:03Z <p>I don´t know about general multi-index Dirichlet series, but there is a good amount of theory on <em>multiple zeta-functions</em> (special cases of what you are asking for). There is plenty of stuff in MathSciNet on this.</p> http://mathoverflow.net/questions/5739/multi-index-dirichlet-series/10798#10798 Answer by Anweshi for multi-index Dirichlet series Anweshi 2010-01-05T11:09:20Z 2010-02-01T19:33:46Z <p>See P. Deligne, <em>Multizeta values</em>, Notes d'exposes, IAS Princeton, for the deep mathematical aspects of this. </p> <p>Also for a general relevance philosophy, see Kontsevich and Zagier, <em>Periods</em>, Mathematics Unlimited(2001). An electronic version is available <a href="http://www.maths.gla.ac.uk/~tl/periods.ps" rel="nofollow">here</a>.</p> <p>There are various references, including those of Zudilin, Cartier, Zagier, Terasoma, Oesterle(On polylogarithms), Manin(iterated integrals and ....). Please look into mathscinet. </p> <p>There seem to be many papers by Dorian Goldfeld and collaborators, too.</p>