Editing Dedekind zeta function

{{short description|Generalization of the Riemann zeta function for algebraic number fields}}
In [[mathematics]], the '''Dedekind zeta function''' of an [[algebraic number field]] ''K'', generally denoted ζ<sub>''K''</sub>(''s''), is a generalization of the [[Riemann zeta function]] (which is obtained in the case where ''K'' is the [[rational number|field of rational numbers]] '''Q'''). It can be defined as a [[Dirichlet series]], it has an [[Euler product]] expansion, it satisfies a [[functional equation (L-function)|functional equation]], it has an [[analytic continuation]] to a [[meromorphic function]] on the [[complex plane]] '''C''' with only a [[simple pole]] at ''s''&nbsp;=&nbsp;1, and its values encode arithmetic data of ''K''. The [[extended Riemann hypothesis]] states that if ''ζ''<sub>''K''</sub>(''s'')&nbsp;=&nbsp;0 and 0&nbsp;<&nbsp;Re(''s'')&nbsp;<&nbsp;1, then Re(''s'')&nbsp;=&nbsp;1/2.

The Dedekind zeta function is named for [[Richard Dedekind]] who introduced it in his supplement to [[Peter Gustav Lejeune Dirichlet]]'s [[Vorlesungen über Zahlentheorie]].<ref>{{harvnb|Narkiewicz|2004|loc=§7.4.1}}</ref>

==Definition and basic properties==
Let ''K'' be an [[algebraic number field]]. Its Dedekind zeta function is first defined for [[complex number]]s ''s'' with [[real part]] Re(''s'')&nbsp;>&nbsp;1 by the Dirichlet series

:<math>\zeta_K (s) = \sum_{I \subseteq \mathcal{O}_K} \frac{1}{(N_{K/\mathbf{Q}} (I))^{s}}</math>

where ''I'' ranges through the non-zero [[ideal (ring theory)|ideals]] of the [[ring of integers]] ''O''<sub>''K''</sub> of ''K'' and ''N''<sub>''K''/'''Q'''</sub>(''I'') denotes the [[absolute norm]] of ''I'' (which is equal to both the [[Index of a subgroup|index]] [''O''<sub>''K''</sub>&nbsp;:&nbsp;''I''] of ''I'' in ''O''<sub>''K''</sub> or equivalently the [[cardinality]] of the [[quotient ring]] ''O''<sub>''K''</sub>&nbsp;/&nbsp;''I''). This sum converges absolutely for all complex numbers ''s'' with [[real part]] Re(''s'')&nbsp;>&nbsp;1. In the case ''K''&nbsp;=&nbsp;'''Q''', this definition reduces to that of the Riemann zeta function.

===Euler product===
The Dedekind zeta function of <math>K</math> has an Euler product which is a product over all the non-zero [[prime ideal]]s <math>\mathfrak{p}</math> of <math>\mathcal{O}_K</math>

:<math>\zeta_K (s) = \prod_{\mathfrak{p} \subseteq \mathcal{O}_K} \frac{1}{1-N_{K/\mathbf{Q}}(\mathfrak{p})^{-s}},\text{ for Re}(s)>1.</math>

This is the expression in analytic terms of the [[Dedekind domain|uniqueness of prime factorization of ideals]] in <math>\mathcal{O}_K</math>. For <math>\mathrm{Re}(s)>1,\ \zeta_K(s)</math> is non-zero.

===Analytic continuation and functional equation===
[[Erich Hecke]] first proved that ''ζ''<sub>''K''</sub>(''s'') has an analytic continuation to a meromorphic function that is analytic at all points of the complex plane except for one simple pole at ''s''&nbsp;=&nbsp;1. The [[Residue (complex analysis)|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&nbsp;&minus;&nbsp;''s''. Specifically, let Δ<sub>''K''</sub> denote the [[Discriminant of an algebraic number field|discriminant]] of ''K'', let ''r''<sub>1</sub> (resp. ''r''<sub>2</sub>) denote the number of real [[Algebraic number theory#Primes and places|places]] (resp. complex places) of ''K'', and let

:<math>\Gamma_\mathbf{R}(s)=\pi^{-s/2}\Gamma(s/2)</math>

and

:<math>\Gamma_\mathbf{C}(s)=
(2\pi)^{-s}\Gamma(s)</math>

where Γ(''s'') is the [[gamma function]]. Then, the functions

:<math>\Lambda_K(s)=\left|\Delta_K\right|^{s/2}\Gamma_\mathbf{R}(s)^{r_1}\Gamma_\mathbf{C}(s)^{r_2}\zeta_K(s)\qquad
\Xi_K(s)=\tfrac12(s^2+\tfrac14)\Lambda_K(\tfrac12+is)
</math>

satisfy the functional equation

:<math>\Lambda_K(s)=\Lambda_K(1-s).\qquad
\Xi_K(-s)=\Xi_K(s)\;</math>

==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 [[class number formula|analytic class number formula]] relates the residue at ''s''&nbsp;=&nbsp;1 to the [[class number (number theory)|class number]] ''h''(''K'') of ''K'', the [[regulator of an algebraic number field|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''&nbsp;=&nbsp;0 where it has a zero whose order ''r'' is equal to the [[rank of an abelian group|rank]] of the unit group of ''O''<sub>''K''</sub> and the leading term is given by

:<math>\lim_{s\rightarrow0}s^{-r}\zeta_K(s)=-\frac{h(K)R(K)}{w(K)}.</math>

It follows from the functional equation that <math>r=r_1+r_2-1</math>.
Combining the functional equation and the fact that Γ(''s'') is infinite at all integers less than or equal to zero yields that ''ζ''<sub>''K''</sub>(''s'') vanishes at all negative even integers. It even vanishes at all negative odd integers unless ''K'' is [[totally real number field|totally real]] (i.e. ''r''<sub>2</sub>&nbsp;=&nbsp;0; e.g. '''Q''' or a [[real quadratic field]]). In the totally real case, [[Carl Ludwig Siegel]] showed that ''ζ''<sub>''K''</sub>(''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-function]]s. For example, when ''K'' is a [[quadratic field]] this shows that the ratio

:<math>\frac{\zeta_K(s)}{\zeta_{\mathbf{Q}}(s)}</math>

is the ''L''-function ''L''(''s'',&nbsp;χ), where χ is a [[Jacobi symbol]] used as [[Dirichlet character]]. 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]] ''G'', its Dedekind zeta function is the [[Artin L-function|Artin ''L''-function]] of the [[regular representation]] of ''G'' and hence has a factorization in terms of Artin ''L''-functions of [[irreducible representation|irreducible]] [[Artin representation]]s of ''G''.

The relation with Artin L-functions shows that if ''L''/''K'' is a Galois extension then <math>\frac{\zeta_L(s)}{\zeta_K(s)}</math> is holomorphic (<math>\zeta_K(s)</math> "divides" <math>\zeta_L(s)</math>): for general extensions the result would follow from the [[Artin conjecture (L-functions)|Artin conjecture for L-functions]].<ref name=Mar19>Martinet (1977) p.19</ref>

Additionally, ''ζ''<sub>''K''</sub>(''s'') is the [[Hasse–Weil zeta function]] of [[Spectrum of a ring|Spec]] ''O''<sub>''K''</sub><ref>{{harvnb|Deninger|1994|loc=§1}}</ref> and the [[motivic L-function|motivic ''L''-function]] of the [[motive (algebraic geometry)|motive]] coming from the [[cohomology]] of Spec ''K''.<ref>{{harvnb|Flach|2004|loc=§1.1}}</ref>

==Arithmetically equivalent fields==

Two fields are called arithmetically equivalent if they have the same Dedekind zeta function. {{harvs|txt | last1=Bosma | first1=Wieb | last2=de Smit | first2=Bart | year=2002 | volume=2369 }} used [[Gassmann triple]]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.

{{harvtxt|Perlis|1977}} showed that two [[number field]]s ''K'' and ''L'' are arithmetically equivalent if and only if all but finitely many prime numbers ''p'' have the same [[splitting_of_prime_ideals_in_Galois_extensions|inertia degree]]s in the two fields, i.e., if <math>\mathfrak p_i</math> are the prime ideals in ''K'' lying over ''p'', then  the tuples <math>(\dim_{\mathbf Z/p} \mathcal O_K / \mathfrak p_i)</math> need to be the same for ''K'' and for ''L'' for almost all ''p''.

==Notes==
{{reflist}}

==References==

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