L-function

L''-functionsL''-functionL-series-function-functionscomplex L-seriesL-functions
In mathematics, an L-function is a meromorphic function on the complex plane, associated to one out of several categories of mathematical objects.wikipedia
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Riemann zeta function

zeta functionzeta functionsRiemann zeta-function
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. We 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.
Many generalizations of the Riemann zeta function, such as Dirichlet series, Dirichlet -functions and -functions, are known.

Selberg class

Selberg series
The Selberg class 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.
In mathematics, the Selberg class is an axiomatic definition of a class of L-functions.

P-adic L-function

p''-adic ''L''-functionsp-adic zeta functionp''-adic ''L''-function
In that case results have been obtained for p-adic L-functions, which describe certain Galois modules.
In mathematics, a p-adic zeta function, or more generally a p-adic L-function, is a function analogous to the Riemann zeta function, or more general L-functions, but whose domain and target are p-adic (where p is a prime number).

Functional equation (L-function)

functional equationfunctional equation of L-functionsfunctional equations
functional equation, with respect to some vertical line Re(s) = constant;
In mathematics, the L-functions of number theory are expected to have several characteristic properties, one of which is that they satisfy certain functional equations. There is an elaborate theory of what these equations should be, much of which is still conjectural.

Algebraic K-theory

algebraic ''K''-theoryK-theoryalgebraic -theory
interesting values at integers related to quantities from algebraic K-theory
The subject also includes classical number-theoretic topics like quadratic reciprocity and embeddings of number fields into the real numbers and complex numbers, as well as more modern concerns like the construction of higher regulators and special values of L-functions.

Peter Swinnerton-Dyer

Sir Peter Swinnerton-DyerSir (Henry) Peter (Francis) Swinnerton-Dyer Bt.Sir (Henry) Peter Francis Swinnerton-Dyer, 16th Baronet
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.
As a mathematician he was best known for his part in the Birch and Swinnerton-Dyer conjecture relating algebraic properties of elliptic curves to special values of L-functions, which was developed with Bryan Birch during the first half of the 1960s with the help of machine computation, and for his work on the Titan operating system.

Dirichlet L-function

Dirichlet ''L''-functionsL-functionsDirichlet ''L''-function
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-function

Random matrix

random matricesrandom matrix theory random matrix theory
The statistics of the zero distributions are of interest because of their connection to problems like the Generalized Riemann hypothesis, distribution of prime numbers, etc. The connections with random matrix theory and quantum chaos are also of interest.
In number theory, the distribution of zeros of the Riemann zeta function (and other L-functions) is modelled by the distribution of eigenvalues of certain random matrices.

Hasse–Weil zeta function

Hasse–Weil ''L''-functionHasse–Weil ''L''-functionsL-series
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.
In mathematics, the Hasse–Weil zeta function attached to an algebraic variety V defined over an algebraic number field K is one of the two most important types of L-function.

Generalized Riemann hypothesis

extended Riemann hypothesisERHGeneralized Riemann Hypothesis (GRH)
Generalized Riemann hypothesis
Various geometrical and arithmetical objects can be described by so-called global L-functions, which are formally similar to the Riemann zeta-function.

Langlands program

geometric Langlands programfunctorialityfurther developed
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 L-functions, were defined several decades earlier, and to L-functions attached to general automorphic representations.
The Artin reciprocity law applies to a Galois extension of an algebraic number field whose Galois group is abelian; it assigns L-functions to the one-dimensional representations of this Galois group, and states that these L-functions are identical to certain Dirichlet L-series or more general series (that is, certain analogues of the Riemann zeta function) constructed from Hecke characters.

Special values of L-functions

Bloch–Kato conjecturesspecial valuesvalues of L-functions
Special values of L-functions
There are two families of conjectures, formulated for general classes of L-functions (the very general setting being for L-functions L(s) associated to Chow motives over number fields), the division into two reflecting the questions of:

Meromorphic function

meromorphicmeromorphic functionsmeromorphically
In mathematics, an L-function is a meromorphic function on the complex plane, associated to one out of several categories of mathematical objects.

Function (mathematics)

functionfunctionsmathematical function
In mathematics, an L-function is a meromorphic function on the complex plane, associated to one out of several categories of mathematical objects.

Complex plane

complex number planeArgand diagramimaginary axis
In mathematics, an L-function is a meromorphic function on the complex plane, associated to one out of several categories of mathematical objects.

Mathematical object

objectsmathematical objectsgeometric object
In mathematics, an L-function is a meromorphic function on the complex plane, associated to one out of several categories of mathematical objects.

Dirichlet series

Dirichlet-Serieszeta
An L-series is a Dirichlet series, usually convergent on a half-plane, that may give rise to an L-function via analytic continuation. We 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. a Dirichlet series, and then by an expansion as an Euler product indexed by prime numbers.

Convergent series

convergenceconvergesconverge
An L-series is a Dirichlet series, usually convergent on a half-plane, that may give rise to an L-function via analytic continuation.

Half-space (geometry)

half-spacehalf-planehalfspace
An L-series is a Dirichlet series, usually convergent on a half-plane, that may give rise to an L-function via analytic continuation.

Analytic continuation

analytically continuedanalytic extensionmeromorphic continuation
An L-series is a Dirichlet series, usually convergent on a half-plane, that may give rise to an L-function via analytic continuation. We 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.

Conjecture

conjecturalconjecturesconjectured
The theory of L-functions has become a very substantial, and still largely conjectural, part of contemporary analytic number theory.

Analytic number theory

analyticanalytic number theoristanalytic techniques
The theory of L-functions has become a very substantial, and still largely conjectural, part of contemporary analytic number theory.

Dirichlet character

conductorcharacterConductor of a Dirichlet character
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.

Infinite set

infiniteinfinitelyinfinitely many
We 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.

Euler product

Euler's product formulaproduct of Euler factorsStephens constant
a Dirichlet series, and then by an expansion as an Euler product indexed by prime numbers.