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Wikipedia / L-function

An L-function is a meromorphic function on the complex plane, and one out of several categories of mathematical objects studied in analytic number theory and related fields. L-functions share fundamental properties and characteristics with the Riemann zeta function, which serves as the prototypical example of an L-function; therefore, L-functions are generalisations of the Riemann zeta function. Some important conjectures involving L-functions are, consequently, the Riemann hypothesis and its generalisations.

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

Fundamental subclasses of L-functions were built on the work of Leonhard Euler (which is now known as the Riemann zeta function). Most notably, the mathematicians Bernhard Riemann (1826–1866), Richard Dedekind (1831–1916), Erich Hecke (1887–1947) and Emil Artin (1898–1962) investigated the subclasses of L-functions, discovering eponymous L-functions each.

The terms "L-function" and "zeta-function" are often used synonymously due to the fundamentally similar and derivative nature of the work, however, not all zeta-functions are L-functions. Most notably, the Prime zeta function is not an L-function, since they cannot be analytically extended to the entire complex plane.

Definition

As one might infer from the introduction, there is still no general and widely accepted definition of an L-function and its construction. Various constructions and definitions as per various prominent authors can be found as follows.

Iwaniec and Kowalski's Analytic Number Theory, 2004

This definition is abstract and incomplete in the sense that it doesn't specify the arithmetic objects to which he assigns L-functions, nor the exact mechanism of this assignment. However, it includes properties generally expected of L-functions.

The definition is extended and starts by defining 6 preliminary definitions, as follows:

Dirichlet Series and Euler Product

is associated with a Dirichlet series:

which is also called an L-series, and an Euler product:

for all natural numbers denotes the set of all prime numbers.

are called local roots or local parameters.

, the expression

th factor in the Euler product, which is called the Euler factor of at.

Gamma Factor

is assigned a so-called gamma factor:

denotes the gamma function denotes the automorphic number at infinity, or at the infinite prime point.

Leader (Conductor)

are called unbranched.

Complete L-Function

, we can now define the so-called complete L-function:

Root

is associated with a complex number

This complex number is called the root.

Dual, Arithmetic Object

is associated with another arithmetic object (which is not specified in detail within the framework of this abstract definition). It is called the dual is also a Dirichlet series

an Euler product

is called self-dual.

Conditions to satisfy

satisfies the definition of an L-function according to Iwaniec and Kowalski:Iwaniec and Kowalski’s definition reflects the fact that a function that is considered an L-function typically appears as a mapping of the L-function to a mathematical object (e.g., a Dirichlet character or an algebraic number field). Their definition is abstract and incomplete, as it leaves open the question of exactly what those mathematical objects are and how that mapping is to be carried out.

Atle Selberg in the Amalfi Conference on Analytic Number Theory, 1992

The definition proposed by the Norwegian-American mathematician Atle Selberg in 1989 is independent of other mathematical objects. In a non-abstract, unambiguous definition, he specifies a subset of the set of all Dirichlet series whose elements must satisfy certain properties: absolute convergence of the Dirichlet series, analytic continuation, functional equation, Ramanujan conjecture and Euler product. This subset is today referred to as the Selberg class.

The overarching hypothesis and the motivating background for the definition of the Selberg class is the so-called Great Riemann Hypothesis. Applied to the Selberg class, this hypothesis states that no zero of an analytic continuation of a Dirichlet series in the Selberg class has a real part greater than 1/2. In the case of the (supposedly) simplest element of the Selberg class (the Riemann Dirichlet series together with its analytic continuation to the Riemann zeta function), this conjecture corresponds to the Riemann Hypothesis, which to this day has been neither proven nor disproven.

It is against this backdrop that the remaining shortcomings in the definition of the term L-function must be viewed: one would like to define the term "L-function" in such a way that L-functions verifiably satisfy the Great Riemann Hypothesis. Yet, on the other hand, one has not even been able to prove the simplest case (the Riemann Hypothesis for the Riemann zeta function), which could be a sign of a lack of understanding of the Riemann zeta function and thus makes it difficult to provide a clear definition of the generalized concept of a L-function.

Examples

This section provides an overview of basic examples of L-functions.

Riemann Zeta Function

The simplest example of an L-function, and at the same time the fundamental basis for any definition of the term L-function based on the work of Leonhard Euler.

in the sense of the definition approach by Iwaniec and Kowalski, to which this L-function can be assigned, is the field of the rational numbers. Its Dirichlet series

that is

. The degree of the Euler product of the Riemann zeta function is

, the following holds:

. Usually, the following gamma factor is used for the Riemann zeta function:

so that the complete Riemann zeta function takes the form

with residues of −1 and 1, respectively., then it satisfies the root number

the functional equation

for the continued Riemann zeta function as well, it satisfies the functional equation

The (analytically extended) Riemann zeta function raises one of the most important questions in analytic number theory: the question of the exact location of its so-called non-trivial zeros. The Riemann Hypothesis. A proof of this hypothesis would allow for particularly accurate estimates of the distribution of prime numbers.

Dirichlet L-functions

coefficients are equal to 1, in Dirichlet L-functions these are defined using a Dirichlet character:

is given, i.e., a group homomorphism from the group of elements that are invertible with respect to multiplication in the residue class ring to the circle group is called primitive is the generator of if it is not already given by a composition

The trivial Dirichlet characters, and 0 otherwise. The trivial Dirichlet character modulo 1 is called the principal character..

as follows: With

the Dirichlet series (also called the Dirichlet L-series)

is absolute.

this also holds for the corresponding Euler product, and we have the identity

. As with the Riemann zeta function,

is called “odd”), then

is also the leader of the Dirichlet L-function:

The complete Dirichlet L-function thus has the form

. This results in an integral function is a non-trivial Dirichlet character. with residue 1., i.e.,

can be determined using the Gaussian sum

denotes the circular number denotes the imaginary unit denotes the exponential function. With

then the extended, complete Dirichlet L-function satisfies the functional equation

. The Dirichlet L-functions include the Riemann zeta function, since it arises from the trivial Dirichlet character modulo 1, i.e., the principal character.

In 1837, the German mathematician Peter Gustav Dirichlet used the Dirichlet L-functions, named after him, to prove the Dirichlet's prime number theorem, according to which in every arithmetic sequence (also called an arithmetic progression)

, there are infinitely many prime numbers. .

Dedekind L-functions

of rational numbers, the simplest algebraic number field. Dedekind L-functions generalize this reference to arbitrary algebraic number fields, i.e., finite field extensions be its degree of extension be its integral domain its discriminant be the number of real embeddings.

The Dedekind L-function (also known as the Dedekind zeta function by

runs through all integer ideals that are distinct from the zero ideal denotes the absolute norm. The coefficients of the Dirichlet series

are thus

, as does the corresponding Euler product

Here, the product extends over all prime ideals, the following identity holds

depend on the decomposition behavior

has a “prime ideal decomposition” that is unique up to the order of the factors

divide is called the , which is called the :

in the identity

in the polynomial ring.

is given by

is the class number is the regulator is the number of unit roots.

Dedekind L-functions always have a root of 1:

satisfies the functional equation

, namely via the definition

. One of its fascinating properties is the so-called analytic class number formula takes the following form:

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, with respect to some vertical line Re(s) = constant;
interesting values at integers related to quantities from algebraic K-theory.

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-functionss, which describe certain Galois modules.

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. The fractal structure of the distributions has been studied using rescaled range analysis. The self-similarity of the zero distribution is quite remarkable, and is characterized by a large fractal dimension of 1.9. This rather large fractal dimension is found over zeros covering at least fifteen orders of magnitude for the Riemann zeta function, and also for the zeros of other L-functions of different orders and conductors.

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 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.