Talk:L-function

Untitled

Historically, what does the L in L-function or L-series stand for? Lagrange? Ninte 11:41, 13 July 2006 (UTC)

The L-function should be defined (either technically or intuitively) and bolded at the start. MP (talk) 19:22, 8 May 2006 (UTC)

A request for clarification

I am not a mathematician myself, and have no idea of what L-functions are. However, from browsing the Wikipedia I managed to understand that there are several classes of such functions: Hecke L-functions, Dirichlet L-functions, Dedekind set functions, etc.. I think much work is needed in all the corresponding articles so that even the layperson is able to have an idea of how they are related. And this is of paramount importance, for there are analogous statements of the Riemann Hypothesis for several of those classes, and this has implications on other conjectures. For example: Goldbach's weak conjecture can be derived from the Generalized Riemann Hypothesis, but the latter needs the Riemann Hypothesis to be proved for Dirichlet L-functions (or so I seem to have understood from the corresponding articles). However, Hecke L-functions were constructed to encompass Dirichlet L-functions. Does this mean that proving RH for Hecke L-functions proves Goldbach's weak conjecture as well? From what I could see in Wikipedia, it seems that the answer is yes, but the leaps made from the word L-function to the word character to the word L-series create some confusion to laypeople like myself. Is it possible to say something of the kind: all Dirichlet L-functions are Hecke L-functions? I believe the general article on L-functions should be made into a departure point for all people who would like to know, even without understanding the mathematics, how concepts are inter-related. A good start would be to name all important classes of L-functions here.

Your request is very meaningful. Unfortunately, however, there are genuine mathematical difficulties involved. Forget about

even the layperson is able to have an idea of how they are related —

even most mathematicians, certainly a very large number outside of number theory, have no idea of what all these different L-function are! Generally speaking, only GRH is interesting for applications to traditional analytic number theory. We know very little about analytical properties of very general L-functions (even the location of the poles is difficult to pin down), and their theory for the most part has quite different flavor. Having said that, let me answer your specific question: yes, a notion of Hecke L-function is a (vast) generalization of the Dirichlet L-function, but not as general as Artin or Langlands L-function. Arcfrk 06:35, 10 March 2007 (UTC)

Some indication that the domain is not always the complex plane, but can be much more general, would be nice (or have I missed it?). This should, of course, be late in the article, but should indicate how the situation changes in other fields. Septentrionalis PMAnderson 22:21, 16 September 2008 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:L-function/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

needs a definition of some L-function Jakob.scholbach 14:13, 2 September 2007 (UTC)

Last edited at 14:13, 2 September 2007 (UTC). Substituted at 02:16, 5 May 2016 (UTC)