In
mathematics, the Dedekind zeta function of an
algebraic number field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a ...
''K'', generally denoted ζ
''K''(''s''), is a generalization of the
Riemann zeta function (which is obtained in the case where ''K'' is the
field of rational numbers Q). It can be defined 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 analyti ...
, it has 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 Leonhar ...
expansion, it satisfies a
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 mea ...
, it has an
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 n ...
to a
meromorphic function
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. ...
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 th ...
C with only a
simple pole
In complex analysis (a branch of mathematics), a pole is a certain type of singularity of a complex-valued function of a complex variable. In some sense, it is the simplest type of singularity. Technically, a point is a pole of a function if i ...
at ''s'' = 1, and its values encode arithmetic data of ''K''. The
extended 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 ...
states that if ''ζ''
''K''(''s'') = 0 and 0 < Re(''s'') < 1, then Re(''s'') = 1/2.
The Dedekind zeta function is named for
Richard Dedekind
Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and
the axiomatic foundations of arithmetic. His ...
who introduced it in his supplement to
Peter Gustav Lejeune Dirichlet
Johann Peter Gustav Lejeune Dirichlet (; 13 February 1805 – 5 May 1859) was a German mathematician who made deep contributions to number theory (including creating the field of analytic number theory), and to the theory of Fourier series and ...
's
Vorlesungen über Zahlentheorie
(German for ''Lectures on Number Theory'') is the name of several different textbooks of number theory. The best known was written by Peter Gustav Lejeune Dirichlet and Richard Dedekind, and published in 1863. Others were written by Leopold Krone ...
.
Definition and basic properties
Let ''K'' be an
algebraic number field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a ...
. Its Dedekind zeta function is first defined for complex numbers ''s'' with
real part
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
Re(''s'') > 1 by the Dirichlet series
:
where ''I'' ranges through the non-zero
ideals
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considered ...
of the
ring of integers
In mathematics, the ring of integers of an algebraic number field K is the ring of all algebraic integers contained in K. An algebraic integer is a root of a monic polynomial with integer coefficients: x^n+c_x^+\cdots+c_0. This ring is often d ...
''O''
''K'' of ''K'' and ''N''
''K''/Q(''I'') denotes the
absolute norm In commutative algebra, the norm of an ideal is a generalization of a norm of an element in the field extension. It is particularly important in number theory since it measures the size of an ideal of a complicated number ring in terms of an ide ...
of ''I'' (which is equal to both the
index
Index (or its plural form indices) may refer to:
Arts, entertainment, and media Fictional entities
* Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index''
* The Index, an item on a Halo megastru ...
''K'' : ''I''">'O''''K'' : ''I''of ''I'' in ''O''
''K'' or equivalently the
cardinality
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
of
quotient ring
In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. I ...
''O''
''K'' / ''I''). This sum converges absolutely for all complex numbers ''s'' with
real part
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
Re(''s'') > 1. In the case ''K'' = Q, this definition reduces to that of the Riemann zeta function.
Euler product
The Dedekind zeta function of
has an Euler product which is a product over all the
prime ideals
of
:
This is the expression in analytic terms of the
uniqueness of prime factorization of ideals in
. For
is non-zero.
Analytic continuation and functional equation
Erich Hecke
Erich Hecke (20 September 1887 – 13 February 1947) was a German mathematician known for his work in number theory and the theory of modular forms.
Biography
Hecke was born in Buk, Province of Posen, German Empire (now Poznań, Poland). He o ...
first proved that ''ζ''
''K''(''s'') has an analytic continuation to the complex plane as a meromorphic function, having a simple pole only at ''s'' = 1. The
residue at that pole is given by the
analytic class number formula In number theory, the class number formula relates many important invariants of a number field to a special value of its Dedekind zeta function.
General statement of the class number formula
We start with the following data:
* is a number field. ...
and is made up of important arithmetic data involving invariants of the
unit group
In algebra, a unit of a ring is an invertible element for the multiplication of the ring. That is, an element of a ring is a unit if there exists in such that
vu = uv = 1,
where is the multiplicative identity; the element is unique for this ...
and
class group
In number theory, the ideal class group (or class group) of an algebraic number field is the quotient group where is the group of fractional ideals of the ring of integers of , and is its subgroup of principal ideals. The class group is a me ...
of ''K''.
The Dedekind zeta function satisfies a functional equation relating its values at ''s'' and 1 − ''s''. Specifically, let Δ
''K'' denote the
discriminant
In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the ori ...
of ''K'', let ''r''
1 (resp. ''r''
2) denote the number of real
places
Place may refer to:
Geography
* Place (United States Census Bureau), defined as any concentration of population
** Census-designated place, a populated area lacking its own municipal government
* "Place", a type of street or road name
** Often ...
(resp. complex places) of ''K'', and let
:
and
:
where Γ(''s'') is the
Gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except th ...
. Then, the functions
:
satisfy the functional equation
:
Special values
Analogously to the Riemann zeta function, the values of the Dedekind zeta function at integers encode (at least conjecturally) important arithmetic data of the field ''K''. For example, the
analytic class number formula In number theory, the class number formula relates many important invariants of a number field to a special value of its Dedekind zeta function.
General statement of the class number formula
We start with the following data:
* is a number field. ...
relates the residue at ''s'' = 1 to the
class number ''h''(''K'') of ''K'', the
regulator ''R''(''K'') of ''K'', the number ''w''(''K'') of roots of unity in ''K'', the absolute discriminant of ''K'', and the number of real and complex places of ''K''. Another example is at ''s'' = 0 where it has a zero whose order ''r'' is equal to the
rank
Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as:
Level or position in a hierarchical organization
* Academic rank
* Diplomatic rank
* Hierarchy
* H ...
of the unit group of ''O''
''K'' and the leading term is given by
:
It follows from the functional equation that
.
Combining the functional equation and the fact that Γ(''s'') is infinite at all integers less than or equal to zero yields that ''ζ''
''K''(''s'') vanishes at all negative even integers. It even vanishes at all negative odd integers unless ''K'' is
totally real
In number theory, a number field ''F'' is called totally real if for each embedding of ''F'' into the complex numbers the image lies inside the real numbers. Equivalent conditions are that ''F'' is generated over Q by one root of an integer polyn ...
(i.e. ''r''
2 = 0; e.g. Q or a
real quadratic field
In algebraic number theory, a quadratic field is an algebraic number field of degree two over \mathbf, the rational numbers.
Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free integer different from 0 a ...
). In the totally real case,
Carl Ludwig Siegel
Carl Ludwig Siegel (31 December 1896 – 4 April 1981) was a German mathematician specialising in analytic number theory. He is known for, amongst other things, his contributions to the Thue–Siegel–Roth theorem in Diophantine approximation ...
showed that ''ζ''
''K''(''s'') is a non-zero rational number at negative odd integers.
Stephen Lichtenbaum
Stephen Lichtenbaum (1939 in Brooklyn) is an American mathematician who is working in the fields of algebraic geometry, algebraic number theory and algebraic K-theory.
Lichtenbaum was an undergraduate at Harvard University (bachelor's degree " ...
conjectured specific values for these rational numbers in terms of the
algebraic K-theory
Algebraic ''K''-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called ''K''-groups. These are groups in the sense ...
of ''K''.
Relations to other ''L''-functions
For the case in which ''K'' is an
abelian extension In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is also cyclic, the extension is also called a cyclic extension. Going in the other direction, a Galois extension is called solvabl ...
of Q, its Dedekind zeta function can be written as a product of
Dirichlet L-function
In mathematics, a Dirichlet ''L''-series is a function of the form
:L(s,\chi) = \sum_^\infty \frac.
where \chi is a Dirichlet character and ''s'' a complex variable with real part greater than 1. It is a special case of a Dirichlet series. ...
s. For example, when ''K'' is a
quadratic field
In algebraic number theory, a quadratic field is an algebraic number field of degree two over \mathbf, the rational numbers.
Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free integer different from 0 ...
this shows that the ratio
:
is the ''L''-function ''L''(''s'', χ), where χ is a
Jacobi symbol
Jacobi symbol for various ''k'' (along top) and ''n'' (along left side). Only are shown, since due to rule (2) below any other ''k'' can be reduced modulo ''n''. Quadratic residues are highlighted in yellow — note that no entry with a J ...
used as
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) \c ...
. That the zeta function of a quadratic field is a product of the Riemann zeta function and a certain Dirichlet ''L''-function is an analytic formulation of the
quadratic reciprocity
In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Due to its subtlety, it has many formulations, but the most standard s ...
law of Gauss.
In general, if ''K'' is a
Galois extension
In mathematics, a Galois extension is an algebraic field extension ''E''/''F'' that is normal and separable; or equivalently, ''E''/''F'' is algebraic, and the field fixed by the automorphism group Aut(''E''/''F'') is precisely the base fiel ...
of Q with
Galois group
In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the po ...
''G'', its Dedekind zeta function is the
Artin ''L''-function of the
regular representation
In mathematics, and in particular the theory of group representations, the regular representation of a group ''G'' is the linear representation afforded by the group action of ''G'' on itself by translation.
One distinguishes the left regular r ...
of ''G'' and hence has a factorization in terms of Artin ''L''-functions of
irreducible Artin representation In mathematics, the Artin conductor is a number or ideal (ring theory), ideal associated to a character of a Galois group of a local field, local or global field, global field (mathematics), field, introduced by as an expression appearing in the fu ...
s of ''G''.
The relation with Artin L-functions shows that if ''L''/''K'' is a Galois extension then
is holomorphic (
"divides"
): for general extensions the result would follow from the
Artin conjecture for L-functions.
[Martinet (1977) p.19]
Additionally, ''ζ''
''K''(''s'') is the
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 reduci ...
of
Spec Spec may refer to:
*Specification (technical standard), an explicit set of requirements to be satisfied by a material, product, or service
**datasheet, or "spec sheet"
People
* Spec Harkness (1887-1952), American professional baseball pitcher
* ...
''O''
''K'' and the
motivic ''L''-function of the
motive coming from the
cohomology
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
of Spec ''K''.
Arithmetically equivalent fields
Two fields are called arithmetically equivalent if they have the same Dedekind zeta function. used
Gassmann triple In mathematics, a Gassmann triple (or Gassmann-Sunada triple) is a group ''G'' together with two faithful actions on sets ''X'' and ''Y'', such that ''X'' and ''Y'' are not isomorphic as ''G''-sets but every element of ''G'' has the same number o ...
s to give some examples of pairs of non-isomorphic fields that are arithmetically equivalent. In particular some of these pairs have different class numbers, so the Dedekind zeta function of a number field does not determine its class number.
showed that two
number field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a ...
s ''K'' and ''L'' are arithmetically equivalent if and only if all but finitely many prime numbers ''p'' have the same
inertia degrees in the two fields, i.e., if
are the prime ideals in ''K'' lying over ''p'', then the tuples
need to be the same for ''K'' and for ''L'' for almost all ''p''.
Notes
References
*
*Section 10.5.1 of
*
*
*
*
*
{{L-functions-footer
Zeta and L-functions
Algebraic number theory