@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:-W90JFV07-V
  skos:prefLabel "multiplicative function"@en, "fonction multiplicative"@fr ;
  a skos:Concept ;
  skos:narrower psr:-CZPGQHT4-T .

psr:-CZPGQHT4-T
  skos:broader psr:-W90JFV07-V, psr:-NHFK3Q1R-H, psr:-ZRKNWDWK-L ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Ramanujan_tau_function>, <https://fr.wikipedia.org/wiki/Fonction_tau_de_Ramanujan> ;
  a skos:Concept ;
  skos:prefLabel "Ramanujan tau function"@en, "fonction tau de Ramanujan"@fr ;
  skos:inScheme psr: ;
  dc:modified "2024-10-18"^^xsd:date ;
  dc:created "2023-08-22"^^xsd:date ;
  skos:definition """The <b>Ramanujan tau function</b>, studied by Ramanujan (1916), is the function <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 \\	au :\\\\mathbb {N} \\
ightarrow \\\\mathbb {Z} }">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>τ<!-- τ --></mi>         <mo>:</mo>         <mrow class="MJX-TeXAtom-ORD">           <mi mathvariant="double-struck">N</mi>         </mrow>         <mo stretchy="false">→<!-- → --></mo>         <mrow class="MJX-TeXAtom-ORD">           <mi mathvariant="double-struck">Z</mi>         </mrow>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\	au :\\\\mathbb {N} \\
ightarrow \\\\mathbb {Z} }</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf647811a6b781caa9aace34ae2966dde50b9fcc" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.982ex; height:2.176ex;" alt="{\\\\displaystyle \\	au :\\\\mathbb {N} \\
ightarrow \\\\mathbb {Z} }"></span> defined by the following identity:  <dl><dd><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 \\\\sum _{n\\\\geq 1}\\	au (n)q^{n}=q\\\\prod _{n\\\\geq 1}\\\\left(1-q^{n}\\
ight)^{24}=q\\\\phi (q)^{24}=\\\\eta (z)^{24}=\\\\Delta (z),}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <munder>           <mo>∑<!-- ∑ --></mo>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>             <mo>≥<!-- ≥ --></mo>             <mn>1</mn>           </mrow>         </munder>         <mi>τ<!-- τ --></mi>         <mo stretchy="false">(</mo>         <mi>n</mi>         <mo stretchy="false">)</mo>         <msup>           <mi>q</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>           </mrow>         </msup>         <mo>=</mo>         <mi>q</mi>         <munder>           <mo>∏<!-- ∏ --></mo>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>             <mo>≥<!-- ≥ --></mo>             <mn>1</mn>           </mrow>         </munder>         <msup>           <mrow>             <mo>(</mo>             <mrow>               <mn>1</mn>               <mo>−<!-- − --></mo>               <msup>                 <mi>q</mi>                 <mrow class="MJX-TeXAtom-ORD">                   <mi>n</mi>                 </mrow>               </msup>             </mrow>             <mo>)</mo>           </mrow>           <mrow class="MJX-TeXAtom-ORD">             <mn>24</mn>           </mrow>         </msup>         <mo>=</mo>         <mi>q</mi>         <mi>ϕ<!-- ϕ --></mi>         <mo stretchy="false">(</mo>         <mi>q</mi>         <msup>           <mo stretchy="false">)</mo>           <mrow class="MJX-TeXAtom-ORD">             <mn>24</mn>           </mrow>         </msup>         <mo>=</mo>         <mi>η<!-- η --></mi>         <mo stretchy="false">(</mo>         <mi>z</mi>         <msup>           <mo stretchy="false">)</mo>           <mrow class="MJX-TeXAtom-ORD">             <mn>24</mn>           </mrow>         </msup>         <mo>=</mo>         <mi mathvariant="normal">Δ<!-- Δ --></mi>         <mo stretchy="false">(</mo>         <mi>z</mi>         <mo stretchy="false">)</mo>         <mo>,</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\sum _{n\\\\geq 1}\\	au (n)q^{n}=q\\\\prod _{n\\\\geq 1}\\\\left(1-q^{n}\\
ight)^{24}=q\\\\phi (q)^{24}=\\\\eta (z)^{24}=\\\\Delta (z),}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e214fc00c33091551a03e7ef0fd782610e3d0734" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:56.39ex; height:5.676ex;" alt="{\\\\displaystyle \\\\sum _{n\\\\geq 1}\\	au (n)q^{n}=q\\\\prod _{n\\\\geq 1}\\\\left(1-q^{n}\\
ight)^{24}=q\\\\phi (q)^{24}=\\\\eta (z)^{24}=\\\\Delta (z),}"></span></dd></dl> where <span class="texhtml"><i>q</i> = exp(2<i>πiz</i>)</span> with <span class="texhtml">Im <i>z</i> &gt; 0</span>, <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 \\\\phi }">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>ϕ<!-- ϕ --></mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\phi }</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\\\\displaystyle \\\\phi }"></span> is the Euler function, <span class="texhtml mvar" style="font-style:italic;">η</span> is the Dedekind eta function, and the function <span class="texhtml">Δ(<i>z</i>)</span> is a holomorphic cusp form of weight 12 and level 1, known as the discriminant modular form (some authors, notably Apostol, write <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 \\\\Delta /(2\\\\pi )^{12}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi mathvariant="normal">Δ<!-- Δ --></mi>         <mrow class="MJX-TeXAtom-ORD">           <mo>/</mo>         </mrow>         <mo stretchy="false">(</mo>         <mn>2</mn>         <mi>π<!-- π --></mi>         <msup>           <mo stretchy="false">)</mo>           <mrow class="MJX-TeXAtom-ORD">             <mn>12</mn>           </mrow>         </msup>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\Delta /(2\\\\pi )^{12}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b407d80631fedaf843dd6e17071dadcf6a2a4591" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.278ex; height:3.176ex;" alt="{\\\\displaystyle \\\\Delta /(2\\\\pi )^{12}}"></span> instead of <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 \\\\Delta }">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi mathvariant="normal">Δ<!-- Δ --></mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\Delta }</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32769037c408874e1890f77554c65f39c523ebe2" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\\\\displaystyle \\\\Delta }"></span>). It appears in connection to an "error term" involved in counting the number of ways of expressing an integer as a sum of 24 squares. A formula due to Ian G. Macdonald was given in Dyson (1972). 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Ramanujan_tau_function">https://en.wikipedia.org/wiki/Ramanujan_tau_function</a>)"""@en, """La <b>fonction tau de Ramanujan</b>, étudiée par Ramanujan, est la fonction <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 \\	au :\\\\mathbb {N} \\
ightarrow \\\\mathbb {Z} }">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>τ<!-- τ --></mi>         <mo>:</mo>         <mrow class="MJX-TeXAtom-ORD">           <mi mathvariant="double-struck">N</mi>         </mrow>         <mo stretchy="false">→<!-- → --></mo>         <mrow class="MJX-TeXAtom-ORD">           <mi mathvariant="double-struck">Z</mi>         </mrow>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\	au :\\\\mathbb {N} \\
ightarrow \\\\mathbb {Z} }</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf647811a6b781caa9aace34ae2966dde50b9fcc" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.982ex; height:2.176ex;" alt="{\\\\displaystyle \\	au :\\\\mathbb {N} \\
ightarrow \\\\mathbb {Z} }"></span> défini par l'identité suivante :  <dl><dd><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 \\\\sum _{n\\\\geq 1}\\	au (n)q^{n}=q\\\\prod _{n\\\\geq 1}\\\\left(1-q^{n}\\
ight)^{24}=q\\\\phi (q)^{24}=\\\\eta (z)^{24}=\\\\Delta (z),}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <munder>           <mo>∑<!-- ∑ --></mo>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>             <mo>≥<!-- ≥ --></mo>             <mn>1</mn>           </mrow>         </munder>         <mi>τ<!-- τ --></mi>         <mo stretchy="false">(</mo>         <mi>n</mi>         <mo stretchy="false">)</mo>         <msup>           <mi>q</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>           </mrow>         </msup>         <mo>=</mo>         <mi>q</mi>         <munder>           <mo>∏<!-- ∏ --></mo>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>             <mo>≥<!-- ≥ --></mo>             <mn>1</mn>           </mrow>         </munder>         <msup>           <mrow>             <mo>(</mo>             <mrow>               <mn>1</mn>               <mo>−<!-- − --></mo>               <msup>                 <mi>q</mi>                 <mrow class="MJX-TeXAtom-ORD">                   <mi>n</mi>                 </mrow>               </msup>             </mrow>             <mo>)</mo>           </mrow>           <mrow class="MJX-TeXAtom-ORD">             <mn>24</mn>           </mrow>         </msup>         <mo>=</mo>         <mi>q</mi>         <mi>ϕ<!-- ϕ --></mi>         <mo stretchy="false">(</mo>         <mi>q</mi>         <msup>           <mo stretchy="false">)</mo>           <mrow class="MJX-TeXAtom-ORD">             <mn>24</mn>           </mrow>         </msup>         <mo>=</mo>         <mi>η<!-- η --></mi>         <mo stretchy="false">(</mo>         <mi>z</mi>         <msup>           <mo stretchy="false">)</mo>           <mrow class="MJX-TeXAtom-ORD">             <mn>24</mn>           </mrow>         </msup>         <mo>=</mo>         <mi mathvariant="normal">Δ<!-- Δ --></mi>         <mo stretchy="false">(</mo>         <mi>z</mi>         <mo stretchy="false">)</mo>         <mo>,</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\sum _{n\\\\geq 1}\\	au (n)q^{n}=q\\\\prod _{n\\\\geq 1}\\\\left(1-q^{n}\\
ight)^{24}=q\\\\phi (q)^{24}=\\\\eta (z)^{24}=\\\\Delta (z),}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e214fc00c33091551a03e7ef0fd782610e3d0734" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:56.39ex; height:5.676ex;" alt="{\\\\displaystyle \\\\sum _{n\\\\geq 1}\\	au (n)q^{n}=q\\\\prod _{n\\\\geq 1}\\\\left(1-q^{n}\\
ight)^{24}=q\\\\phi (q)^{24}=\\\\eta (z)^{24}=\\\\Delta (z),}"></span></dd></dl> où <span class="texhtml"><i>q</i> = exp(2<i>πiz</i>)</span> avec <span class="texhtml">Im <i>z</i> &gt; 0</span>, <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 \\\\phi }">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>ϕ<!-- ϕ --></mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\phi }</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\\\\displaystyle \\\\phi }"></span> est l'indicatrice d'Euler, <span class="texhtml mvar" style="font-style:italic;">η</span> est la fonction êta de Dedekind, et la fonction <span class="texhtml">Δ(<i>z</i>)</span> est une forme parabolique de poids 12 et de niveau 1, connue sous le nom de forme modulaire discriminant. Elle apparaît être en relation avec un « terme d'erreur » impliqué dans le comptage du nombre de façons d'exprimer un entier comme une somme de 24 carrés. Une formule due à Ian G. Macdonald a été donnée dans Dyson 1972. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Fonction_tau_de_Ramanujan">https://fr.wikipedia.org/wiki/Fonction_tau_de_Ramanujan</a>)"""@fr ;
  skos:narrower psr:-R6KBTF6L-5 .

psr:-NHFK3Q1R-H
  skos:prefLabel "fonction L"@fr, "L-function"@en ;
  a skos:Concept ;
  skos:narrower psr:-CZPGQHT4-T .

psr:-R6KBTF6L-5
  skos:prefLabel "Ramanujan conjecture"@en, "conjecture de Ramanujan"@fr ;
  a skos:Concept ;
  skos:broader psr:-CZPGQHT4-T .

psr: a skos:ConceptScheme .
psr:-ZRKNWDWK-L
  skos:prefLabel "forme modulaire"@fr, "modular form"@en ;
  a skos:Concept ;
  skos:narrower psr:-CZPGQHT4-T .