L-function

The theory of L-functions is a topical area of ​​analytic number theory, in which generalizations of the Riemann ζ - function and Dirichlet L- series studied. Here, their general properties are given with systematic methods, where one is, however, dependent on the initial guesses, that is, one is often of a rigorous proof yet removed.

L-functions

Right at the beginning is aware between L- series distinguished ( for example, the Dirichlet series for the Riemann ζ - function) and L-functions. The latter are functions in the complex plane figures and by various methods - produced from the ranks - mainly by analytic continuation.

Thus, the design starts with an L- series, which is first converted into an Eulerian product, which is indicated with primes. It then follows the proof that this product converges in a " right half-plane " of. It then attempts to continue the so- defined function in a very analytical ( possibly with additional poles ).

The so- defined meromorphic - or as a meromorphic suspected - continuation is called L- function. But already classic examples teach that might useful information can be included in points for which the L- series does not converge.

The term L- function includes definitely known examples such as the ζ - functions. The so-called Selberg class S is based on a set of axioms gained for L-functions. Thereby, the investigations are systematized by rather than individual functions, the associated class is studied.

Assumed properties

One can choose from well-known examples see which properties should have a theory of L-functions, ie they should

Detailed investigations have revealed a large number of plausible conjectures generated, for example on the exact type of the currently specified functional equations. Since the Riemann ζ - function related by their values ​​at even positive integers ( odd and negative values ​​) with the Bernoulli numbers, it is natural to look for a generalization of the Bernoulli numbers in the given theory. We used the body of the p- adic numbers, making certain Galois modules are described.

The statistical properties of the zeros distribution of L- functions are different therefore of interest because it is related to general problems, for example with a hypothesis about the distribution of prime numbers and with other so-called generalized Riemann hypothesis. The connection with the theories of random matrices and the so-called quantum chaos is also of interest. The fractal structure of the distributions was also investigated with so-called scale analysis .. The self-similarity of the zeros distribution is very remarkable, and is characterized by a large value of the fractal dimension ~ 1.9. This very high value is for more than 15 orders of magnitude of the zeros of the Riemann ζ - distribution function and also for the zeros of other L-functions.

The conjecture of Birch and Swinnerton - Dyer

→ Main article: conjecture of Birch and Swinnerton - Dyer

One of the more important problems, both for the history of generalized L-functions and in terms of many still unresolved issues is given by a presumption which was erected in the early 1960s by Bryan Birch and Peter Swinnerton - Dyer.

This assumption is based on an elliptic curve E. The problem to be solved is to predict the rank of E over the set of rational numbers (or any other number fields ): So it is for E the number of free generators of the group their rational points are determined.

Much of the research in this area was unified by the point of view to achieve a better knowledge of the L-functions. This was a paradigmatic goal of the new resulting theory of L-functions.

The general theory

This development led within a few years to the Langlands program, for which it is to some extent complementary. Langlands investigations relate largely to Artinian L-functions, which, as the Hecke L-functions have been defined several decades earlier, and L-functions associated with general automorphic representations.

Was so clear Gradually, the sense in which the ζ - functions of Hasse and Weil on the same L- functions perform as that of Riemann. It was recognized that some form of analysis, a kind of automorphic analysis, had to be provided. The general case united conceptually to a variety of research programs.

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