@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:-NHFK3Q1R-H
  skos:prefLabel "fonction L"@fr, "L-function"@en ;
  a skos:Concept ;
  skos:narrower psr:-Z00WNCP8-4 .

psr:-R15183XZ-N
  skos:prefLabel "automorphic form"@en, "forme automorphe"@fr ;
  a skos:Concept ;
  skos:narrower psr:-Z00WNCP8-4 .

psr: a skos:ConceptScheme .
psr:-Z00WNCP8-4
  skos:broader psr:-R15183XZ-N, psr:-NHFK3Q1R-H, psr:-ZRKNWDWK-L ;
  skos:altLabel "conjecture de Ramanujan généralisée"@fr, "generalized Ramanujan conjecture"@en ;
  skos:prefLabel "Ramanujan-Petersson conjecture"@en, "conjecture de Ramanujan-Petersson"@fr ;
  dc:modified "2024-10-18"^^xsd:date ;
  dc:created "2023-08-22"^^xsd:date ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Ramanujan%E2%80%93Petersson_conjecture>, <https://fr.wikipedia.org/wiki/Conjecture_de_Ramanujan> ;
  skos:definition """In mathematics, the <b>Ramanujan conjecture</b>, due to Srinivasa Ramanujan (1916, p. 176), states that Ramanujan's tau function given by the Fourier coefficients <span class="texhtml"><i>τ</i>(<i>n</i>)</span> of the cusp form <span class="texhtml">Δ(<i>z</i>)</span> of weight <span class="texhtml">12</span>  <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 \\\\Delta (z)=\\\\sum _{n>0}\\	au (n)q^{n}=q\\\\prod _{n>0}\\\\left(1-q^{n}\\
ight)^{24}=q-24q^{2}+252q^{3}-1472q^{4}+4830q^{5}-\\\\cdots ,}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi mathvariant="normal">Δ<!-- Δ --></mi>         <mo stretchy="false">(</mo>         <mi>z</mi>         <mo stretchy="false">)</mo>         <mo>=</mo>         <munder>           <mo>∑<!-- ∑ --></mo>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>             <mo>&gt;</mo>             <mn>0</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>&gt;</mo>             <mn>0</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>         <mo>−<!-- − --></mo>         <mn>24</mn>         <msup>           <mi>q</mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>2</mn>           </mrow>         </msup>         <mo>+</mo>         <mn>252</mn>         <msup>           <mi>q</mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>3</mn>           </mrow>         </msup>         <mo>−<!-- − --></mo>         <mn>1472</mn>         <msup>           <mi>q</mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>4</mn>           </mrow>         </msup>         <mo>+</mo>         <mn>4830</mn>         <msup>           <mi>q</mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>5</mn>           </mrow>         </msup>         <mo>−<!-- − --></mo>         <mo>⋯<!-- ⋯ --></mo>         <mo>,</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\Delta (z)=\\\\sum _{n&gt;0}\\	au (n)q^{n}=q\\\\prod _{n&gt;0}\\\\left(1-q^{n}\\
ight)^{24}=q-24q^{2}+252q^{3}-1472q^{4}+4830q^{5}-\\\\cdots ,}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6533c181df1c619eecea8158e02a48559746a0e0" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:82.167ex; height:5.509ex;" alt="{\\\\displaystyle \\\\Delta (z)=\\\\sum _{n>0}\\	au (n)q^{n}=q\\\\prod _{n>0}\\\\left(1-q^{n}\\
ight)^{24}=q-24q^{2}+252q^{3}-1472q^{4}+4830q^{5}-\\\\cdots ,}"></span></dd></dl> where <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 q=e^{2\\\\pi iz}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>q</mi>         <mo>=</mo>         <msup>           <mi>e</mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>2</mn>             <mi>π<!-- π --></mi>             <mi>i</mi>             <mi>z</mi>           </mrow>         </msup>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle q=e^{2\\\\pi iz}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbd4a07533fb952a40f930a40b1d7437aad86b26" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.584ex; height:3.009ex;" alt="{\\\\displaystyle q=e^{2\\\\pi iz}}"></span>, satisfies  <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 |\\	au (p)|\\\\leq 2p^{11/2},}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mrow class="MJX-TeXAtom-ORD">           <mo stretchy="false">|</mo>         </mrow>         <mi>τ<!-- τ --></mi>         <mo stretchy="false">(</mo>         <mi>p</mi>         <mo stretchy="false">)</mo>         <mrow class="MJX-TeXAtom-ORD">           <mo stretchy="false">|</mo>         </mrow>         <mo>≤<!-- ≤ --></mo>         <mn>2</mn>         <msup>           <mi>p</mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>11</mn>             <mrow class="MJX-TeXAtom-ORD">               <mo>/</mo>             </mrow>             <mn>2</mn>           </mrow>         </msup>         <mo>,</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle |\\	au (p)|\\\\leq 2p^{11/2},}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/133a353a3478ee8cb9656eaab1ab56991b74d1e6" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.072ex; height:3.343ex;" alt="{\\\\displaystyle |\\	au (p)|\\\\leq 2p^{11/2},}"></span></dd></dl> when <span class="texhtml mvar" style="font-style:italic;">p</span> is a prime number. The <b>generalized Ramanujan conjecture</b> or <b>Ramanujan–Petersson conjecture</b>, introduced by Petersson (1930), is a generalization to other modular forms or automorphic forms. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Ramanujan%E2%80%93Petersson_conjecture">https://en.wikipedia.org/wiki/Ramanujan%E2%80%93Petersson_conjecture</a>)"""@en, """En mathématiques, la conjecture de Ramanujan, due à Srinivasa Ramanujan (et démontrée par Pierre Deligne en 1973), prédit certaines propriétés arithmétiques ainsi que le comportement asymptotique de la fonction tau qu'il a définie. La conjecture de Ramanujan généralisée, ou conjecture de Ramanujan-Petersson, introduite par Hans Petersson en 1930, en est une généralisation à d'autres formes modulaires ou automorphes. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Conjecture_de_Ramanujan">https://fr.wikipedia.org/wiki/Conjecture_de_Ramanujan</a>)"""@fr ;
  a skos:Concept ;
  skos:inScheme psr: .

psr:-ZRKNWDWK-L
  skos:prefLabel "forme modulaire"@fr, "modular form"@en ;
  a skos:Concept ;
  skos:narrower psr:-Z00WNCP8-4 .