@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .
@prefix dc: <http://purl.org/dc/terms/> .
@prefix xsd: <http://www.w3.org/2001/XMLSchema#> .

psr:-MK8TB7ZV-N
  skos:prefLabel "Dirichlet series"@en, "série de Dirichlet"@fr ;
  a skos:Concept ;
  skos:narrower psr:-CJGHSPXZ-9 .

psr: a skos:ConceptScheme .
psr:-CJGHSPXZ-9
  skos:exactMatch <https://en.wikipedia.org/wiki/Dedekind_zeta_function>, <https://fr.wikipedia.org/wiki/Fonction_z%C3%AAta_de_Dedekind> ;
  skos:definition """In mathematics, the <b>Dedekind zeta function</b> of an algebraic number field <i>K</i>, generally denoted ζ<sub><i>K</i></sub>(<i>s</i>), is a generalization of the Riemann zeta function (which is obtained in the case where <i>K</i> is the field of rational numbers <b>Q</b>). It can be defined as a Dirichlet series, it has an Euler product expansion, it satisfies a functional equation, it has an analytic continuation to a meromorphic function on the complex plane <b>C</b> with only a simple pole at <i>s</i>&nbsp;=&nbsp;1, and its values encode arithmetic data of <i>K</i>. The extended Riemann hypothesis states that if <i>ζ</i><sub><i>K</i></sub>(<i>s</i>)&nbsp;=&nbsp;0 and 0&nbsp;&lt;&nbsp;Re(<i>s</i>)&nbsp;&lt;&nbsp;1, then Re(<i>s</i>)&nbsp;=&nbsp;1/2. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Dedekind_zeta_function">https://en.wikipedia.org/wiki/Dedekind_zeta_function</a>)"""@en, """En mathématiques, la <b>fonction zêta de Dedekind</b> est une série de Dirichlet définie pour tout corps de nombres <span class="texhtml"><i>K</i></span>. C'est la fonction de la variable complexe <span class="texhtml"><i>s</i></span> définie par la somme infinie&nbsp;:
<br/>
<br/><center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\zeta _{K}(s)=\\\\sum \\\\left(N_{K/\\\\mathbb {Q} }(I)\\
ight)^{-s}~({\\	ext{si Re}}(s)>1)}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>ζ<!-- ζ --></mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>K</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>s</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <mo>∑<!-- ∑ --></mo>
<br/>        <msup>
<br/>          <mrow>
<br/>            <mo>(</mo>
<br/>            <mrow>
<br/>              <msub>
<br/>                <mi>N</mi>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mi>K</mi>
<br/>                  <mrow class="MJX-TeXAtom-ORD">
<br/>                    <mo>/</mo>
<br/>                  </mrow>
<br/>                  <mrow class="MJX-TeXAtom-ORD">
<br/>                    <mi mathvariant="double-struck">Q</mi>
<br/>                  </mrow>
<br/>                </mrow>
<br/>              </msub>
<br/>              <mo stretchy="false">(</mo>
<br/>              <mi>I</mi>
<br/>              <mo stretchy="false">)</mo>
<br/>            </mrow>
<br/>            <mo>)</mo>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mo>−<!-- − --></mo>
<br/>            <mi>s</mi>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mtext>&nbsp;</mtext>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mtext>si Re</mtext>
<br/>        </mrow>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>s</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>&gt;</mo>
<br/>        <mn>1</mn>
<br/>        <mo stretchy="false">)</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\zeta _{K}(s)=\\\\sum \\\\left(N_{K/\\\\mathbb {Q} }(I)\\
ight)^{-s}~({\\	ext{si Re}}(s)&gt;1)}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1c6f2b6a56271b7bc715df991a5d9fad5a70646" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.338ex; width:39.942ex; height:3.843ex;" alt="{\\\\displaystyle \\\\zeta _{K}(s)=\\\\sum \\\\left(N_{K/\\\\mathbb {Q} }(I)\\
ight)^{-s}~({\\	ext{si Re}}(s)>1)}"></span></center>
<br/>prise sur tous les idéaux <span class="texhtml"><i>I</i></span> non nuls de l'anneau <span class="texhtml"><i>O<sub>K</sub></i></span> des entiers de <span class="texhtml"><i>K</i></span>, où <span class="texhtml"><i>N</i><sub><i>K</i>/ℚ</sub>(<i>I</i>)</span> désigne la norme de <span class="texhtml"><i>I</i></span> (relative au corps ℚ des rationnels). Cette norme est égale au cardinal de l'anneau quotient <span class="texhtml"><i>O<sub>K</sub></i>/<i>I</i></span>. En particulier, <span class="texhtml">ζ<sub>ℚ</sub></span> est la fonction zêta de Riemann. Les propriétés de la fonction méromorphe <span class="texhtml">ζ<sub><i>K</i></sub></span> ont une signification considérable en théorie algébrique des nombres. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Fonction_z%C3%AAta_de_Dedekind">https://fr.wikipedia.org/wiki/Fonction_z%C3%AAta_de_Dedekind</a>)"""@fr ;
  skos:prefLabel "fonction zêta de Dedekind"@fr, "Dedekind zeta function"@en ;
  dc:created "2023-08-04"^^xsd:date ;
  a skos:Concept ;
  skos:broader psr:-MK8TB7ZV-N ;
  dc:modified "2023-08-04"^^xsd:date ;
  skos:inScheme psr: .