In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 f ...
''K'', generally denoted ζ
''K''(''s''), is a generalization of 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) > ...
(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 analy ...
, 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 Leonhard Eu ...
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 meaning ...
, 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 ...
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. The ...
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 ...
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 ...
at ''s'' = 1, and its values encode arithmetic data of ''K''. The
extended Riemann hypothesis 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'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 f ...
. Its Dedekind zeta function is first defined for complex numbers ''s'' with
real part Re(''s'') > 1 by the Dirichlet series
:
where ''I'' ranges through the non-zero
ideals 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 deno ...
''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 idea ...
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 ''O''
''K'' / ''I''). This sum converges absolutely for all complex numbers ''s'' with
real part 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 ideal
In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together wi ...
s
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 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 and is made up of important arithmetic data involving invariants of the
unit group and
class group 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 orig ...
of ''K'', let ''r''
1 (resp. ''r''
2) denote the number of real
places (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 excep ...
. 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 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 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 (i.e. ''r''
2 = 0; e.g. Q or a
real quadratic field). In the totally real case,
Carl Ludwig Siegel showed that ''ζ''
''K''(''s'') is a non-zero rational number at negative odd integers.
Stephen Lichtenbaum conjectured specific values for these rational numbers in terms of the
algebraic K-theory of ''K''.
Relations to other ''L''-functions
For the case in which ''K'' is an
abelian extension of Q, its Dedekind zeta function can be written as a product of
Dirichlet L-functions. For example, when ''K'' is a
quadratic field 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 ...
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) \ch ...
. 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 law of Gauss.
In general, if ''K'' is a
Galois extension 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 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 associated to a character of a Galois group of a local or global field, introduced by as an expression appearing in the functional equation of an Artin L-function.
Local Artin conductors
...
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 of
Spec ''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 view ...
of Spec ''K''.
Arithmetically equivalent fields
Two fields are called arithmetically equivalent if they have the same Dedekind zeta function. used
Gassmann triples 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 fields ''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
*
*
*
*
*
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Zeta and L-functions
Algebraic number theory