@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:-GQMC7X75-1
  skos:prefLabel "Odlyzko-Schönhage algorithm"@en, "algorithme d'Odlyzko-Schönhage"@fr ;
  skos:definition """En mathématiques, l'<b>algorithme d'Odlyzko-Schönhage</b> est un algorithme d'évaluation rapide de la fonction zêta de Riemann   <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 \\\\zeta ~:~z\\\\mapsto \\\\sum _{n=1}^{+\\\\infty }{\\rac {1}{n^{z}}}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>ζ<!-- ζ --></mi>         <mtext> </mtext>         <mo>:</mo>         <mtext> </mtext>         <mi>z</mi>         <mo stretchy="false">↦<!-- ↦ --></mo>         <munderover>           <mo>∑<!-- ∑ --></mo>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>             <mo>=</mo>             <mn>1</mn>           </mrow>           <mrow class="MJX-TeXAtom-ORD">             <mo>+</mo>             <mi mathvariant="normal">∞<!-- ∞ --></mi>           </mrow>         </munderover>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mn>1</mn>             <msup>               <mi>n</mi>               <mrow class="MJX-TeXAtom-ORD">                 <mi>z</mi>               </mrow>             </msup>           </mfrac>         </mrow>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\zeta ~:~z\\\\mapsto \\\\sum _{n=1}^{+\\\\infty }{\\rac {1}{n^{z}}}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60ef6704e7b1444d2e8fdbe4271a083f8167b58a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.87ex; height:7.176ex;" alt="{\\\\displaystyle \\\\zeta ~:~z\\\\mapsto \\\\sum _{n=1}^{+\\\\infty }{\\rac {1}{n^{z}}}}"></span>.</dd></dl> Cet algorithme </span>, présenté en 1988 par  Andrew Odlyzko et Arnold Schönhage, a servi au premier auteur</span> dans le calcul du 10<sup>20</sup> ème zéro et de valeurs proches de la fonction zêta de Riemann, dans le cadre de la vérification de la conjecture connue sous le nom d'hypothèse de Riemann.  
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Algorithme_d%27Odlyzko-Sch%C3%B6nhage">https://fr.wikipedia.org/wiki/Algorithme_d%27Odlyzko-Sch%C3%B6nhage</a>)"""@fr, """In mathematics, the <b>Odlyzko–Schönhage algorithm</b> is a fast algorithm for evaluating the Riemann zeta function at many points, introduced by  (Odlyzko &amp;  Schönhage 1988).  The main point is the use of the fast Fourier transform to speed up the evaluation of a finite Dirichlet series of length <i>N</i> at O(<i>N</i>) equally spaced values from O(<i>N</i><sup>2</sup>) to O(<i>N</i><sup>1+ε</sup>) steps (at the cost of storing O(<i>N</i><sup>1+ε</sup>) intermediate values). The Riemann–Siegel formula  used for  calculating the Riemann zeta function  with imaginary part <i>T</i> uses a finite Dirichlet series with about <i>N</i> = <i>T</i><sup>1/2</sup> terms, so when finding about <i>N</i> values of the Riemann zeta function it is sped up by a factor of about <i>T</i><sup>1/2</sup>.  This reduces the time to find the zeros of the zeta function with imaginary part at most <i>T</i> from  about <i>T</i><sup>3/2+ε</sup> steps  to about <i>T</i><sup>1+ε</sup> steps.  The algorithm can be used not just for the Riemann zeta function, but also for many other functions given by Dirichlet series.  The algorithm was used by Gourdon (2004) to verify the Riemann hypothesis for the first 10<sup>13</sup> zeros of the zeta function. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Odlyzko%E2%80%93Sch%C3%B6nhage_algorithm">https://en.wikipedia.org/wiki/Odlyzko%E2%80%93Sch%C3%B6nhage_algorithm</a>)"""@en ;
  dc:modified "2024-10-18"^^xsd:date ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Odlyzko%E2%80%93Sch%C3%B6nhage_algorithm>, <https://fr.wikipedia.org/wiki/Algorithme_d%27Odlyzko-Sch%C3%B6nhage> ;
  skos:inScheme psr: ;
  dc:created "2023-08-04"^^xsd:date ;
  skos:broader psr:-JR45XVX3-B, psr:-P36V4MHV-V ;
  a skos:Concept .

psr:-P36V4MHV-V
  skos:prefLabel "fonction zêta de Riemann"@fr, "Riemann zeta function"@en ;
  a skos:Concept ;
  skos:narrower psr:-GQMC7X75-1 .

psr:-JR45XVX3-B
  skos:prefLabel "computational number theory"@en, "théorie calculatoire des nombres"@fr ;
  a skos:Concept ;
  skos:narrower psr:-GQMC7X75-1 .

psr: a skos:ConceptScheme .