Function L

The theory of the functions L became a very substantial branch, and still very largely conjecturelle, contemporary Théorie of the numbers. In this part, broad generalizations of the Function Zeta of Riemann and the series L for a Caractère of Dirichlet are built, and their general properties, which in the majority of the cases are out of reach demonstration, are exposed in a systematic way.

The functions L

As in the case of the very known examples, we can distinguish the representation of series (for example infinite series for the Zeta function of Riemann), and the function in the complex plan which is its analytical prolongation. General constructions start with a series L, initially definite like an infinite product, indexed by prime numbers, then by extension like a series of Dirichlet. Estimates are necessary to show that this converges in the higher right part of the plan of the complex numbers.

It is then judicious to conjecture a prolongation méromorphe in the complex plan, as a function L . In the traditional cases, one knows that the useful information is contained in the values and the knowledge of the function L at the points where the series L itself is not a valid representation. The general term of function L includes/understands many known types of Zeta functions. The Classe of Selberg is an attempt to axiomatize the properties of the functions L and to rather encourage the study of the properties common to all these functions than each function L as a single object.

Examples of functions L

the function ζ of Riemann, which is the most traditional example;
the functions L associated with the modular forms via the Transformation with Mellin;
the functions L associated with the characters, which in particular make it possible to show the Théorème of Dirichlet on the Prime numbers in the arithmetic progressions;
the functions L of the reason S

Conjecturelle information

One can list the characteristics of the known examples of functions L which one would wish to see generalized:

localization of the zeros and the poles;
functional equation (function L), with the vertical line respect certain Re ( S ) = constant;
interesting values with the whole values.

A detailed work produced a large body of plausible conjectures, for example in connection with the exact type of equation functional calculuses which could apply. As the Zeta function of Riemann connects its values to the even entireties of the Nombres of Bernoulli, one can consider a suitable generalization of this phenomenon. In this case, results were obtained for what one calls the p-adic functions L, which describe some modules of Welshman.

The example of the conjecture of Birch and Swinnerton-Dyer

See the principal article Conjecture of Birch and Swinnerton-Dyer

One of the most influential examples, and for the history of the most general functions L and for the search for still opened problems, is the conjecture developed by Bryan Birch and Peter Swinnerton-Dyer in the first part of the years 1960. It applies to a elliptic Courbe E, and the problem which it tries to solve is the prediction of the row of an elliptic curve on the whole of the rational numbers: c.a.d. the number of free generators of its group of rational points. Several preceding work of this field started to be unified around a better knowledge of the functions L. This was something like an example of paradigm of the theory incipient from the functions L.

Extent of the general theory

This development preceded the Programme by Langlands of a few years, and can be looked like its complementary: the work of Langlands is largely related to the functions L of Artin, which, like those of Hecke, were defined several decades earlier.

Gradually it became clearer in which direction the construction of Hasse-Weil could be made to work to provide functions L valid, in the analytical direction: it must exist certain entries starting from the analysis, which wants to say analysis automorphe . The general case now unifies on a conceptual level a number different of research programs.

Some bonds to go further:

References

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