L-function

The theory of "L"-functions has become a very substantial, and still largely conjectural, part of contemporary number theory. In it, broad generalisations of the Riemann zeta function and the "L"-series for a Dirichlet character are constructed, and their general properties, in most cases still out of reach of proof, are set out in a systematic way.

"L"-functions

We should distinguish at the outset between the L-series, an infinite series representation (for example the Dirichlet series for the Riemann zeta-function), and the "L"-function, the function in the complex plane that is its analytic continuation. The general constructions start with an "L"-series, defined first asa Dirichlet series, and then by an expansion as an Euler product indexed by prime numbers.Estimates are required to prove that this converges in some right half-plane of the complex numbers. Then one asks whetherthe function so defined can be analytically continued to the rest of the complex plane (perhaps with some poles).

It is this (possibly conjectural) meromorphic continuation to the complex plane which is called an "L"-function. In the classical cases, already, one knows that useful information is contained in the values and behaviour of the "L"-function at points where the series representation does not converge. The general term "L"-function here includes many known types of zeta-functions. The Selberg class S is an attempt to capture the core properties of "L"-functions in a set of axioms, thus encouraging the study of the properties of the class rather than of individual functions.

Conjectural information

One can list characteristics of known examples of "L"-functions that one would wish to see generalized:

* location of zeros and poles;
* functional equation ("L"-function), with respect to some vertical line Re ("s") = constant;
* interesting values at integers.

Detailed work has produced a large body of plausible conjectures, for example about the exact type of functional equation that should apply. Since the Riemann zeta-function connects through its values at positive even integers (and negative odd integers) to the Bernoulli numbers, one looks for an appropriate generalisation of that phenomenon. In that case results have been obtained for "p"-adic "L"-functions, which describe certain Galois modules.

The example of the Birch and Swinnerton-Dyer conjecture

"See main article Birch and Swinnerton-Dyer conjecture"

One of the influential examples, both for the history of the more general "L"-functions and as a still-open research problem, is the conjecture developed by Bryan Birch and Peter Swinnerton-Dyer in the early part of the 1960s. It applies to an elliptic curve "E", and the problem it attempts to solve is the prediction of the rank of the elliptic curve over the rational numbers (or another global field): i.e. the number of free generators of its group of rational points. Much previous work in the area began to be unified around a better knowledge of "L"-functions. This was something like a paradigm example of the nascent theory of "L"-functions.

Rise of the general theory

This development preceded the Langlands program by a few years, and can be regarded as complementary to it: Langlands' work relates largely to Artin "L"-functions, which, like Hecke's "L"-functions, were defined several decades earlier, and to "L"-functions attached to general automorphic representations.

Gradually it became clearer in what sense the construction of Hasse-Weil zeta-functions might be made to work to provide valid "L"-functions, in the analytic sense: there should be some input from analysis, which meant "automorphic" analysis. The general case now unifies at a conceptual level a number of different research programs.

ee also

*Generalized Riemann hypothesis
*modularity theorem
*Artin conjecture

References

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External links

* [http://www.physorg.com/news124636003.html Glimpses of a new (mathematical) world] - a breakthrough third degree transcendental L-function revealed, "Physorg.com", March 13, 2008
* [http://www.sciencenews.org/view/generic/id/9542/title/Math_Trek__Creeping_Up_on_Riemann Creeping Up on Riemann] , Science News, April 2, 2008
* [http://www.physorg.com/news137248087.html Hunting the elusive L-function]