L-function
Encyclopedia
The theory of L-functions has become a very substantial, and still largely conjectural, part of contemporary analytic 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.
. The general constructions start with an L-series, defined first as
a 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 whether
the function so defined can be analytically continued to the rest of the complex plane (perhaps with some poles).
It is this (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.
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-function
s, which describe certain Galois module
s.
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.
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.
by a few years, and can be regarded as complementary to it: Langlands' work relates largely to Artin L-function
s, 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-function
s 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.
Analytic number theory
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Dirichlet's introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic...
. In it, broad generalisations of the Riemann zeta function and the L-series for a Dirichlet character
Dirichlet character
In number theory, Dirichlet characters are certain arithmetic functions which arise from completely multiplicative characters on the units of \mathbb Z / k \mathbb Z...
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
Analytic continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which...
. The general constructions start with an L-series, defined first as
a Dirichlet series, and then by
an expansion as an Euler product
Euler product
In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers. The name arose from the case of the Riemann zeta-function, where such a product representation was proved by Leonhard Euler.-Definition:...
indexed by prime numbers.
Estimates are required to prove that this converges in some right half-plane of the complex numbers. Then one asks whether
the function so defined can be analytically continued to the rest of the complex plane (perhaps with some poles).
It is this (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)Functional equation (L-function)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...
, 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-function
P-adic L-function
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...
s, which describe certain Galois module
Galois module
In mathematics, a Galois module is a G-module where G is the Galois group of some extension of fields. The term Galois representation is frequently used when the G-module is a vector space over a field or a free module over a ring, but can also be used as a synonym for G-module...
s.
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
Fractal dimension
In fractal geometry, the fractal dimension, D, is a statistical quantity that gives an indication of how completely a fractal appears to fill space, as one zooms down to finer and finer scales. There are many specific definitions of fractal dimension. The most important theoretical fractal...
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.
The example of the Birch and Swinnerton-Dyer conjecture
- See main article Birch and Swinnerton-Dyer conjectureBirch and Swinnerton-Dyer conjectureIn mathematics, the Birch and Swinnerton-Dyer conjecture is an open problem in the field of number theory. Its status as one of the most challenging mathematical questions has become widely recognized; the conjecture was chosen as one of the seven Millennium Prize Problems listed by the Clay...
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
Peter Swinnerton-Dyer
Sir Henry Peter Francis Swinnerton-Dyer, 16th Baronet KBE FRS , commonly known as Peter Swinnerton-Dyer, is an English mathematician specialising in number theory at University of Cambridge...
in the early part of the 1960s. It applies to an elliptic curve
Elliptic curve
In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...
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
Global field
In mathematics, the term global field refers to either of the following:*an algebraic number field, i.e., a finite extension of Q, or*a global function field, i.e., the function field of an algebraic curve over a finite field, equivalently, a finite extension of Fq, the field of rational functions...
): 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 programLanglands program
The Langlands program is a web of far-reaching and influential conjectures that relate Galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles. It was proposed by ....
by a few years, and can be regarded as complementary to it: Langlands' work relates largely to Artin L-function
Artin L-function
In mathematics, an Artin L-function is a type of Dirichlet series associated to a linear representation ρ of a Galois group G. These functions were introduced in the 1923 by Emil Artin, in connection with his research into class field theory. Their fundamental properties, in particular the Artin...
s, 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-function
Hasse-Weil zeta function
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. Such L-functions are called 'global', in that they are defined as Euler products in terms of local zeta functions...
s 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.
See also
- Generalized Riemann hypothesisGeneralized Riemann hypothesisThe Riemann hypothesis is one of the most important conjectures in mathematics. It is a statement about the zeros of the Riemann zeta function. Various geometrical and arithmetical objects can be described by so-called global L-functions, which are formally similar to the Riemann zeta-function...
- Dirichlet L-function
- Modularity theorem
- Artin conjecture
- Special values of L-functionsSpecial values of L-functionsIn mathematics, the study of special values of L-functions is a subfield of number theory devoted to generalising formulae such as the Leibniz formula for pi, namely...
External links
- Glimpses of a new (mathematical) world - a breakthrough third degree transcendental L-function revealed, Physorg.com, March 13, 2008
- Creeping Up on Riemann, Science News, April 2, 2008
- Hunting the elusive L-function