![Riemann-Zeta-Func](https://upload.wikimedia.org/wikipedia/commons/6/65/Riemann-Zeta-Func.png)
In mathematics, an ''L''-function is a
meromorphic
In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are poles of the function. The ...
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
on the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
, associated to one out of several categories of
mathematical object
A mathematical object is an abstract concept arising in mathematics.
In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deductive reasoning and mathematical p ...
s. An ''L''-series is a
Dirichlet series
In mathematics, a Dirichlet series is any series of the form
\sum_^\infty \frac,
where ''s'' is complex, and a_n is a complex sequence. It is a special case of general Dirichlet series.
Dirichlet series play a variety of important roles in analy ...
, usually
convergent on a
half-plane, that may give rise to an ''L''-function via
analytic continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new ...
. The
Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
is an example of an ''L''-function, and one important conjecture involving ''L''-functions is the
Riemann hypothesis
In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in ...
and its
generalization
A generalization is a form of abstraction whereby common properties of specific instances are formulated as general concepts or claims. Generalizations posit the existence of a domain or set of elements, as well as one or more common characteri ...
.
The theory of ''L''-functions has become a very substantial, and still largely
conjectural
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in ...
, part of contemporary
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 Peter Gustav Lejeune Dirichlet's 1837 introduction of Diric ...
. In it, broad generalisations of the Riemann zeta function and the
''L''-series for a
Dirichlet character
In analytic number theory and related branches of mathematics, a complex-valued arithmetic function \chi:\mathbb\rightarrow\mathbb is a Dirichlet character of modulus m (where m is a positive integer) if for all integers a and b:
:1) \chi ...
are constructed, and their general properties, in most cases still out of reach of proof, are set out in a systematic way. Because of the
Euler product formula there is a deep connection between ''L''-functions and the theory of
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
s.
The mathematical field that studies L-functions is sometimes called analytic theory of L-functions.
Construction
We distinguish at the outset between the ''L''-series, an
infinite
Infinite may refer to:
Mathematics
* Infinite set, a set that is not a finite set
*Infinity, an abstract concept describing something without any limit
Music
*Infinite (group), a South Korean boy band
*''Infinite'' (EP), debut EP of American m ...
series representation (for example the
Dirichlet series
In mathematics, a Dirichlet series is any series of the form
\sum_^\infty \frac,
where ''s'' is complex, and a_n is a complex sequence. It is a special case of general Dirichlet series.
Dirichlet series play a variety of important roles in analy ...
for the
Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
), and the ''L''-function, the function in the complex plane that is its
analytic continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new ...
. The general constructions start with an ''L''-series, defined first as a
Dirichlet series
In mathematics, a Dirichlet series is any series of the form
\sum_^\infty \frac,
where ''s'' is complex, and a_n is a complex sequence. It is a special case of general Dirichlet series.
Dirichlet series play a variety of important roles in analy ...
, and then by an expansion as an
Euler product In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers. The original such product was given for the sum of all positive integers raised to a certain power as proven by Leonhard Eu ...
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
pole
Pole may refer to:
Astronomy
*Celestial pole, the projection of the planet Earth's axis of rotation onto the celestial sphere; also applies to the axis of rotation of other planets
*Pole star, a visible star that is approximately aligned with the ...
s).
It is this (conjectural)
meromorphic
In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are poles of the function. The ...
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
In mathematics, the Selberg class is an axiomatic definition of a class of ''L''-functions. The members of the class are Dirichlet series which obey four axioms that seem to capture the essential properties satisfied by most functions that are co ...
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
In mathematics, a functional equation
is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning ...
, 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
In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, ...
, one looks for an appropriate generalisation of that phenomenon. In that case results have been obtained for
''p''-adic ''L''-functions, which describe certain
Galois module
In mathematics, a Galois module is a ''G''-module, with ''G'' being 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 ...
s.
The statistics of the
zero distribution
0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usual ...
s are of interest because of their connection to problems like the generalized Riemann hypothesis, distribution of prime numbers, etc. The connections with
random matrix
In probability theory and mathematical physics, a random matrix is a matrix-valued random variable—that is, a matrix in which some or all elements are random variables. Many important properties of physical systems can be represented mathemat ...
theory and
quantum chaos
Quantum chaos is a branch of physics which studies how chaos theory, chaotic classical dynamical systems can be described in terms of quantum theory. The primary question that quantum chaos seeks to answer is: "What is the relationship betwee ...
are also of interest. The fractal structure of the distributions has been studied using
rescaled range analysis.
The
self-similarity
__NOTOC__
In mathematics, a self-similar object is exactly or approximately similar to a part of itself (i.e., the whole has the same shape as one or more of the parts). Many objects in the real world, such as coastlines, are statistically se ...
of the zero distribution is quite remarkable, and is characterized by a large
fractal dimension
In mathematics, more specifically in fractal geometry, a fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern (strictly speaking, a fractal pattern) changes with the scale at which it is me ...
of 1.9. This rather large fractal dimension is found over zeros covering at least fifteen orders of magnitude for the
Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
, 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
Bryan John Birch FRS (born 25 September 1931) is a British mathematician. His name has been given to the Birch and Swinnerton-Dyer conjecture.
Biography
Bryan John Birch was born in Burton-on-Trent, the son of Arthur Jack and Mary Edith Birch. ...
and
Peter Swinnerton-Dyer
Sir Henry Peter Francis Swinnerton-Dyer, 16th Baronet, (2 August 1927 – 26 December 2018) was an English mathematician specialising in number theory at the University of Cambridge. As a mathematician he was best known for his part in th ...
in the early part of the 1960s. It applies to an
elliptic curve
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...
''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 In mathematics, a global field is one of two type of fields (the other one is local field) which are characterized using valuations. There are two kinds of global fields:
* Algebraic number field: A finite extension of \mathbb
*Global function fi ...
): 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
In representation theory and algebraic number theory, the Langlands program is a web of far-reaching and influential conjectures about connections between number theory and geometry. Proposed by , it seeks to relate Galois groups in algebraic num ...
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 representation
In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group ''G'' to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup \Gamma \subset G of ...
s.
Gradually it became clearer in what sense the construction of
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 a meromorphic function on the complex plane defined in terms of the number of points on the variety after reduc ...
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 hypothesis
The 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, whic ...
*
Dirichlet ''L''-function
*
Automorphic ''L''-function
*
Modularity theorem
The modularity theorem (formerly called the Taniyama–Shimura conjecture, Taniyama-Weil conjecture or modularity conjecture for elliptic curves) states that elliptic curves over the field of rational numbers are related to modular forms. And ...
*
Artin conjecture
*
Special values of ''L''-functions
*
Shimizu ''L''-function
References
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External links
*
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;Articles about a breakthrough third degree transcendental L-function
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