@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:-HTTMJFF9-W
  skos:inScheme psr: ;
  skos:definition """The <b>Erdős–Borwein constant</b> is the sum of the reciprocals of the Mersenne numbers. It is named after Paul Erdős and Peter Borwein. By definition it is:  <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 E=\\\\sum _{n=1}^{\\\\infty }{\\rac {1}{2^{n}-1}}\\\\approx 1.606695152415291763\\\\dots }">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>E</mi>         <mo>=</mo>         <munderover>           <mo>∑<!-- ∑ --></mo>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>             <mo>=</mo>             <mn>1</mn>           </mrow>           <mrow class="MJX-TeXAtom-ORD">             <mi mathvariant="normal">∞<!-- ∞ --></mi>           </mrow>         </munderover>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mn>1</mn>             <mrow>               <msup>                 <mn>2</mn>                 <mrow class="MJX-TeXAtom-ORD">                   <mi>n</mi>                 </mrow>               </msup>               <mo>−<!-- − --></mo>               <mn>1</mn>             </mrow>           </mfrac>         </mrow>         <mo>≈<!-- ≈ --></mo>         <mn>1.606695152415291763</mn>         <mo>…<!-- … --></mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle E=\\\\sum _{n=1}^{\\\\infty }{\\rac {1}{2^{n}-1}}\\\\approx 1.606695152415291763\\\\dots }</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2298257d455e21d9e532da9571a03a6fe9857b84" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:44.778ex; height:6.843ex;" alt="E=\\\\sum _{{n=1}}^{{\\\\infty }}{\\rac  {1}{2^{n}-1}}\\\\approx 1.606695152415291763\\\\dots "> 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Borwein_constant">https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Borwein_constant</a>)"""@en, """La <b>constante d'Erdős-Borwein</b> est la somme <i>E</i> des inverses des nombres de Mersenne (non nécessairement premiers) :  <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 E=\\\\sum _{n=1}^{\\\\infty }{\\rac {1}{2^{n}-1}}\\\\approx 1,606695}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>E</mi>         <mo>=</mo>         <munderover>           <mo>∑<!-- ∑ --></mo>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>             <mo>=</mo>             <mn>1</mn>           </mrow>           <mrow class="MJX-TeXAtom-ORD">             <mi mathvariant="normal">∞<!-- ∞ --></mi>           </mrow>         </munderover>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mn>1</mn>             <mrow>               <msup>                 <mn>2</mn>                 <mrow class="MJX-TeXAtom-ORD">                   <mi>n</mi>                 </mrow>               </msup>               <mo>−<!-- − --></mo>               <mn>1</mn>             </mrow>           </mfrac>         </mrow>         <mo>≈<!-- ≈ --></mo>         <mn>1</mn>         <mo>,</mo>         <mn>606695</mn>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle E=\\\\sum _{n=1}^{\\\\infty }{\\rac {1}{2^{n}-1}}\\\\approx 1,606695}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6dfa8cbd4d1467990b32ccd0897dbb93b968d647" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:28.106ex; height:6.843ex;" alt="{\\\\displaystyle E=\\\\sum _{n=1}^{\\\\infty }{\\rac {1}{2^{n}-1}}\\\\approx 1,606695}"></span> </span>.</dd></dl> On peut démontrer que la première égalité ci-dessus équivaut à chacune des suivantes :  <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 E=\\\\sum _{n=1}^{\\\\infty }{\\rac {1}{2^{n^{2}}}}{\\rac {2^{n}+1}{2^{n}-1}}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>E</mi>         <mo>=</mo>         <munderover>           <mo>∑<!-- ∑ --></mo>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>             <mo>=</mo>             <mn>1</mn>           </mrow>           <mrow class="MJX-TeXAtom-ORD">             <mi mathvariant="normal">∞<!-- ∞ --></mi>           </mrow>         </munderover>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mn>1</mn>             <msup>               <mn>2</mn>               <mrow class="MJX-TeXAtom-ORD">                 <msup>                   <mi>n</mi>                   <mrow class="MJX-TeXAtom-ORD">                     <mn>2</mn>                   </mrow>                 </msup>               </mrow>             </msup>           </mfrac>         </mrow>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mrow>               <msup>                 <mn>2</mn>                 <mrow class="MJX-TeXAtom-ORD">                   <mi>n</mi>                 </mrow>               </msup>               <mo>+</mo>               <mn>1</mn>             </mrow>             <mrow>               <msup>                 <mn>2</mn>                 <mrow class="MJX-TeXAtom-ORD">                   <mi>n</mi>                 </mrow>               </msup>               <mo>−<!-- − --></mo>               <mn>1</mn>             </mrow>           </mfrac>         </mrow>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle E=\\\\sum _{n=1}^{\\\\infty }{\\rac {1}{2^{n^{2}}}}{\\rac {2^{n}+1}{2^{n}-1}}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d9a27738eeaeffc13df8870383f8d4fc64e124f" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:19.885ex; height:6.843ex;" alt="{\\\\displaystyle E=\\\\sum _{n=1}^{\\\\infty }{\\rac {1}{2^{n^{2}}}}{\\rac {2^{n}+1}{2^{n}-1}}}"></span></dd></dl> <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 E=\\\\sum _{m=1}^{\\\\infty }\\\\sum _{n=1}^{\\\\infty }{\\rac {1}{2^{mn}}}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>E</mi>         <mo>=</mo>         <munderover>           <mo>∑<!-- ∑ --></mo>           <mrow class="MJX-TeXAtom-ORD">             <mi>m</mi>             <mo>=</mo>             <mn>1</mn>           </mrow>           <mrow class="MJX-TeXAtom-ORD">             <mi mathvariant="normal">∞<!-- ∞ --></mi>           </mrow>         </munderover>         <munderover>           <mo>∑<!-- ∑ --></mo>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>             <mo>=</mo>             <mn>1</mn>           </mrow>           <mrow class="MJX-TeXAtom-ORD">             <mi mathvariant="normal">∞<!-- ∞ --></mi>           </mrow>         </munderover>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mn>1</mn>             <msup>               <mn>2</mn>               <mrow class="MJX-TeXAtom-ORD">                 <mi>m</mi>                 <mi>n</mi>               </mrow>             </msup>           </mfrac>         </mrow>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle E=\\\\sum _{m=1}^{\\\\infty }\\\\sum _{n=1}^{\\\\infty }{\\rac {1}{2^{mn}}}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57286f8df2e4ea42ce01f5ee538d7b203e3b919c" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:17.206ex; height:6.843ex;" alt="{\\\\displaystyle E=\\\\sum _{m=1}^{\\\\infty }\\\\sum _{n=1}^{\\\\infty }{\\rac {1}{2^{mn}}}}"></span></dd></dl> <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 E=1+\\\\sum _{n=1}^{\\\\infty }{\\rac {1}{2^{n}(2^{n}-1)}}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>E</mi>         <mo>=</mo>         <mn>1</mn>         <mo>+</mo>         <munderover>           <mo>∑<!-- ∑ --></mo>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>             <mo>=</mo>             <mn>1</mn>           </mrow>           <mrow class="MJX-TeXAtom-ORD">             <mi mathvariant="normal">∞<!-- ∞ --></mi>           </mrow>         </munderover>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mn>1</mn>             <mrow>               <msup>                 <mn>2</mn>                 <mrow class="MJX-TeXAtom-ORD">                   <mi>n</mi>                 </mrow>               </msup>               <mo stretchy="false">(</mo>               <msup>                 <mn>2</mn>                 <mrow class="MJX-TeXAtom-ORD">                   <mi>n</mi>                 </mrow>               </msup>               <mo>−<!-- − --></mo>               <mn>1</mn>               <mo stretchy="false">)</mo>             </mrow>           </mfrac>         </mrow>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle E=1+\\\\sum _{n=1}^{\\\\infty }{\\rac {1}{2^{n}(2^{n}-1)}}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9aa6a6e7bb67e6d15761a913956437f02538e10b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:24.029ex; height:6.843ex;" alt="{\\\\displaystyle E=1+\\\\sum _{n=1}^{\\\\infty }{\\rac {1}{2^{n}(2^{n}-1)}}}"></span></dd></dl> <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 E=\\\\sum _{n=1}^{\\\\infty }{\\rac {\\\\sigma _{0}(n)}{2^{n}}}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>E</mi>         <mo>=</mo>         <munderover>           <mo>∑<!-- ∑ --></mo>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>             <mo>=</mo>             <mn>1</mn>           </mrow>           <mrow class="MJX-TeXAtom-ORD">             <mi mathvariant="normal">∞<!-- ∞ --></mi>           </mrow>         </munderover>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mrow>               <msub>                 <mi>σ<!-- σ --></mi>                 <mrow class="MJX-TeXAtom-ORD">                   <mn>0</mn>                 </mrow>               </msub>               <mo stretchy="false">(</mo>               <mi>n</mi>               <mo stretchy="false">)</mo>             </mrow>             <msup>               <mn>2</mn>               <mrow class="MJX-TeXAtom-ORD">                 <mi>n</mi>               </mrow>             </msup>           </mfrac>         </mrow>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle E=\\\\sum _{n=1}^{\\\\infty }{\\rac {\\\\sigma _{0}(n)}{2^{n}}}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7abaa2d993eedc2ae4b9064f74a1b419204e598" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.038ex; height:6.843ex;" alt="{\\\\displaystyle E=\\\\sum _{n=1}^{\\\\infty }{\\rac {\\\\sigma _{0}(n)}{2^{n}}}}"></span></dd></dl> où σ<sub>0</sub> = d est la fonction nombre de diviseurs, une fonction multiplicative donnant le nombre de diviseurs positifs du nombre de départ. Pour démontrer que ces sommes sont égales, notons qu'elles prennent toutes la forme d'une série de Lambert et peuvent ainsi être resommées comme telles.  Paul Erdős a démontré en 1948 que <i>E</i> est un nombre irrationnel</span>. En 1991, Peter Borwein a montré</span> que plus généralement, pour tout entier relatif <i>q</i> et tout rationnel non nul <i>r</i>, <span style="display: block; margin-left:1.6em;"><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=1}^{\\\\infty }{\\rac {1}{q^{n}-r}}\\
otin \\\\mathbb {Q} }">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <munderover>           <mo>∑<!-- ∑ --></mo>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>             <mo>=</mo>             <mn>1</mn>           </mrow>           <mrow class="MJX-TeXAtom-ORD">             <mi mathvariant="normal">∞<!-- ∞ --></mi>           </mrow>         </munderover>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mn>1</mn>             <mrow>               <msup>                 <mi>q</mi>                 <mrow class="MJX-TeXAtom-ORD">                   <mi>n</mi>                 </mrow>               </msup>               <mo>−<!-- − --></mo>               <mi>r</mi>             </mrow>           </mfrac>         </mrow>         <mo>∉<!-- ∉ --></mo>         <mrow class="MJX-TeXAtom-ORD">           <mi mathvariant="double-struck">Q</mi>         </mrow>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\sum _{n=1}^{\\\\infty }{\\rac {1}{q^{n}-r}}\\
otin \\\\mathbb {Q} }</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ce6af731cf1b482fd8526b96dc144c01a27edd1" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.414ex; height:6.843ex;" alt="{\\\\displaystyle \\\\sum _{n=1}^{\\\\infty }{\\rac {1}{q^{n}-r}}\\
otin \\\\mathbb {Q} }"></span></span> dès que la série converge, c'est-à-dire <i>q</i> différent de 0 et ±1 et <i>r</i> non puissance de <i>q</i>.  
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Constante_d%27Erd%C5%91s-Borwein">https://fr.wikipedia.org/wiki/Constante_d%27Erd%C5%91s-Borwein</a>)"""@fr ;
  dc:modified "2024-10-18"^^xsd:date ;
  skos:broader psr:-WX8H0134-J, psr:-RBFVN7DN-2 ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Borwein_constant>, <https://fr.wikipedia.org/wiki/Constante_d%27Erd%C5%91s-Borwein> ;
  a skos:Concept ;
  dc:created "2023-08-03"^^xsd:date ;
  skos:prefLabel "constante d'Erdős-Borwein"@fr, "Erdős-Borwein constant"@en .

psr: a skos:ConceptScheme .
psr:-WX8H0134-J
  skos:prefLabel "nombre irrationnel"@fr, "irrational number"@en ;
  a skos:Concept ;
  skos:narrower psr:-HTTMJFF9-W .

psr:-RBFVN7DN-2
  skos:prefLabel "mathematical constant"@en, "constante mathématique"@fr ;
  a skos:Concept ;
  skos:narrower psr:-HTTMJFF9-W .