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| 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 by specializing to the case where ''K'' is the [[rational number]]s '''Q'''. In particular, 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'' = 1, and its values encode arithmetic data of ''K''. The [[extended Riemann hypothesis]] states that if ''ζ''<sub>''K''</sub>(''s'') = 0 and 0 < Re(''s'') < 1, then Re(''s'') = 1/2.
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| The Dedekind zeta function is named for [[Richard Dedekind]] who introduced them in his supplement to [[Peter Gustav Lejeune Dirichlet]]'s [[Vorlesungen über Zahlentheorie]].<ref>{{harvnb|Narkiewicz|2004|loc=§7.4.1}}</ref>
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| ==Definition and basic properties==
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| Let ''K'' be an [[algebraic number field]]. Its Dedekind zeta function is first defined for complex numbers ''s'' with [[real part]] Re(''s'') > 1 by the Dirichlet series
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| :<math>\zeta_K (s) = \sum_{I \subseteq \mathcal{O}_K} \frac{1}{(N_{K/\mathbf{Q}} (I))^{s}}</math> | | '''MathML''' |
| | :<math forcemathmode="mathml">E=mc^2</math> |
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| 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> : ''I''] of ''I'' in ''O''<sub>''K''</sub> or equivalently the [[cardinality]] of [[quotient ring]] ''O''<sub>''K''</sub> / ''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.
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| | :<math forcemathmode="png">E=mc^2</math> |
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| ===Euler product===
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| The Dedekind zeta function of ''K'' has an Euler product which is a product over all the [[prime ideal]]s ''P'' of ''O''<sub>''K''</sub>
| | :<math forcemathmode="source">E=mc^2</math> --> |
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| :<math>\zeta_K (s) = \prod_{P \subseteq \mathcal{O}_K} \frac{1}{1 - (N_{K/\mathbf{Q}}(P))^{-s}},\text{ for Re}(s)>1.</math> | | <span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples]. |
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| This is the expression in analytic terms of the [[Dedekind domain|uniqueness of prime factorization of the ideals]] ''I'' in ''O''<sub>''K''</sub>. The fact that, for Re(''s'') > 1, ζ<sub>''K''</sub>(''s'') is given by a product of non-zero numbers implies that it is non-zero in this region.
| | ==Demos== |
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| ===Analytic continuation and functional equation===
| | Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]: |
| [[Erich Hecke]] first proved that ''ζ''<sub>''K''</sub>(''s'') has an analytic continuation to the complex plane as a meromorphic function, having a simple pole only at ''s'' = 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''. | |
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| The Dedekind zeta function satisfies a functional equation relating its values at ''s'' and 1 − ''s''. Specifically, let Δ<sub>''K''</sub> denote [[Discriminant of an algebraic number field|discriminant]] of ''K'', let ''r''<sub>1</sub> (resp. ''r''<sub>2</sub>) denote the number of [[real place]]s (resp. [[complex place]]s) of ''K'', and let
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| :<math>\Gamma_\mathbf{R}(s)=\pi^{-s/2}\Gamma(s/2)</math> | | * accessibility: |
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| :<math>\Gamma_\mathbf{C}(s)=2(2\pi)^{-s}\Gamma(s)</math> | | To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages: |
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| where Γ(''s'') is the [[Gamma function]]. Then, the function
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| :<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)</math>
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| satisfies the functional equation
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| :<math>\Lambda_K(s)=\Lambda_K(1-s).\;</math>
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| ==Special values==
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| 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 (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'' = 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
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| :<math>\lim_{s\rightarrow0}s^{-r}\zeta_K(s)=-\frac{h(K)R(K)}{w(K)}.</math>
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| 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> = 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''.
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| ==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
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| :<math>\frac{\zeta_K(s)}{\zeta_{\mathbf{Q}}(s)}</math>
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| is the ''L''-function ''L''(''s'', 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.
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| 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''.
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| 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>
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| 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>
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| ==Arithmetically equivalent fields==
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| 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.
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *{{Citation | last1=Bosma | first1=Wieb | last2=de Smit | first2=Bart | editor1-last=Kohel | editor1-first=David R. | editor2-last=Fieker | editor2-first=Claus | title=Algorithmic number theory (Sydney, 2002) | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Comput. Sci. | isbn=978-3-540-43863-2 | doi=10.1007/3-540-45455-1_6 | mr=2041074 | year=2002 | volume=2369 | chapter=On arithmetically equivalent number fields of small degree | pages=67–79}}
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| *Section 10.5.1 of {{Citation
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| | last=Cohen
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| | first=Henri
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| | author-link=Henri Cohen (number theorist)
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| | title=Number theory, Volume II: Analytic and modern tools
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| | publisher=Springer
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| | location=New York
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| | series=[[Graduate Texts in Mathematics]]
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| | volume=240
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| | year=2007
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| | isbn=978-0-387-49893-5
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| | mr=2312338
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| | doi=10.1007/978-0-387-49894-2
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| }}
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| *{{Citation
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| | last=Deninger
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| | first=Christopher
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| | contribution=''L''-functions of mixed motives
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| | title=Motives, Part 1
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| | series=Proceedings of Symposia in Pure Mathematics
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| | publisher=[[American Mathematical Society]]
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| | volume=55.1
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| | year=1994
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| | editor3-link=Jean-Pierre Serre
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| | isbn=978-0-8218-1635-6
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| | url=http://wwwmath.uni-muenster.de/u/deninger/about/publikat/cd22.ps
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| }}
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| *{{Citation
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| | last=Flach
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| | first=Mathias
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| | contribution=The equivariant Tamagawa number conjecture: a survey
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| | url=http://www.math.caltech.edu/papers/baltimore-final.pdf
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| | title=Stark's conjectures: recent work and new directions
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| | publisher=[[American Mathematical Society]]
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| | series=Contemporary Mathematics
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| | volume=358
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| | pages=79–125
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| | isbn=978-0-8218-3480-0
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| | editor-last=Burns
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| | editor-first=David
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| | editor2-last=Popescu
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| | editor2-first=Christian
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| | editor3-last=Sands
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| | editor3-first=Jonathan
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| | editor4-last=Solomon
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| | editor4-first=David
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| }}
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| *{{citation | last=Martinet | first=J. | chapter=Character theory and Artin L-functions | pages=1-87 | title=Algebraic Number Fields, Proc. Symp. London Math. Soc., Univ. Durham 1975 | editor1-last=Fröhlich | editor1-first=A. | editor1-link=Albrecht Fröhlich | publisher=Academic Press | year=1977 | isbn=0-12-268960-7 | zbl=0359.12015 }}
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| *{{Citation
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| | last=Narkiewicz
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| | first=Władysław
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| | title=Elementary and analytic theory of algebraic numbers
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| | edition=3 | at=Chapter 7
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| | year=2004
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| | publisher=Springer-Verlag
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| | location=Berlin
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| | series=Springer Monographs in Mathematics
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| | isbn=978-3-540-21902-6
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| | mr=2078267
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| }}
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| {{L-functions-footer}}
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| [[Category:Zeta and L-functions]]
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| [[Category:Algebraic number theory]]
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