@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:-VHDD6KJX-8
  skos:prefLabel "analytic number theory"@en, "théorie analytique des nombres"@fr ;
  a skos:Concept ;
  skos:narrower psr:-KN0XMX72-K .

psr:-KN0XMX72-K
  skos:prefLabel "divisor function"@en, "fonction somme des puissances k-ièmes des diviseurs"@fr ;
  skos:broader psr:-W90JFV07-V, psr:-VHDD6KJX-8 ;
  dc:modified "2024-10-18"^^xsd:date ;
  skos:definition """En mathématiques, la <b>fonction "somme des puissances <i>k</i>-ièmes des diviseurs</b>", notée <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 \\\\sigma _{k}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msub>           <mi>σ<!-- σ --></mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>k</mi>           </mrow>         </msub>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\sigma _{k}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/330415aef0a06818b843c55814fda645b793811b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.416ex; height:2.009ex;" alt="\\\\sigma _{k}"></span>, est la fonction multiplicative qui à tout entier <span class="texhtml"><i>n</i> &gt; 0</span> associe la somme des  puissances <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 k}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>k</mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle k}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="k"></span>-ièmes des diviseurs positifs de <span class="texhtml"><i>n</i></span>, où <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 k}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>k</mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle k}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="k"></span> est un nombre complexe quelconque </span>:<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 \\\\sigma _{k}(n)=\\\\sum _{d|n}d^{k}.}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msub>           <mi>σ<!-- σ --></mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>k</mi>           </mrow>         </msub>         <mo stretchy="false">(</mo>         <mi>n</mi>         <mo stretchy="false">)</mo>         <mo>=</mo>         <munder>           <mo>∑<!-- ∑ --></mo>           <mrow class="MJX-TeXAtom-ORD">             <mi>d</mi>             <mrow class="MJX-TeXAtom-ORD">               <mo stretchy="false">|</mo>             </mrow>             <mi>n</mi>           </mrow>         </munder>         <msup>           <mi>d</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>k</mi>           </mrow>         </msup>         <mo>.</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\sigma _{k}(n)=\\\\sum _{d|n}d^{k}.}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc8603c78a7aa25be50fdabfcc3139be4f9e1f88" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:15.414ex; height:6.009ex;" alt="{\\\\displaystyle \\\\sigma _{k}(n)=\\\\sum _{d|n}d^{k}.}"> </center>
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Fonction_somme_des_puissances_k-i%C3%A8mes_des_diviseurs">https://fr.wikipedia.org/wiki/Fonction_somme_des_puissances_k-i%C3%A8mes_des_diviseurs</a>)"""@fr, """In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as the divisor function, it counts the number of divisors of an integer (including 1 and the number itself). It appears in a number of remarkable identities, including relationships on the Riemann zeta function and the Eisenstein series of modular forms. Divisor functions were studied by Ramanujan, who gave a number of important congruences and identities; these are treated separately in the article Ramanujan's sum. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Divisor_function">https://en.wikipedia.org/wiki/Divisor_function</a>)"""@en ;
  skos:inScheme psr: ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Divisor_function>, <https://fr.wikipedia.org/wiki/Fonction_somme_des_puissances_k-i%C3%A8mes_des_diviseurs> ;
  a skos:Concept ;
  dc:created "2023-07-26"^^xsd:date .

psr: a skos:ConceptScheme .
psr:-W90JFV07-V
  skos:prefLabel "multiplicative function"@en, "fonction multiplicative"@fr ;
  a skos:Concept ;
  skos:narrower psr:-KN0XMX72-K .