@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: a skos:ConceptScheme .
psr:-MWBTPGLM-B
  skos:prefLabel "piecewise function"@en, "fonction affine par morceaux"@fr ;
  a skos:Concept ;
  skos:narrower psr:-DT18GRDQ-T .

psr:-DT18GRDQ-T
  skos:exactMatch <https://en.wikipedia.org/wiki/Chebyshev_function>, <https://fr.wikipedia.org/wiki/Fonction_de_Tchebychev> ;
  dc:modified "2024-10-18"^^xsd:date ;
  skos:definition """En mathématiques, la <b>fonction de Tchebychev</b> peut désigner deux fonctions utilisées en théorie des nombres. La <b>première fonction de Tchebychev</b> <span class="texhtml"><i>ϑ</i>(<i>x</i>)</span> ou <span class="texhtml"><i>θ</i>(<i>x</i>)</span> est donnée par  <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 \\\\vartheta (x)=\\\\sum _{p\\\\leq x}\\\\ln p}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>ϑ<!-- ϑ --></mi>         <mo stretchy="false">(</mo>         <mi>x</mi>         <mo stretchy="false">)</mo>         <mo>=</mo>         <munder>           <mo>∑<!-- ∑ --></mo>           <mrow class="MJX-TeXAtom-ORD">             <mi>p</mi>             <mo>≤<!-- ≤ --></mo>             <mi>x</mi>           </mrow>         </munder>         <mi>ln</mi>         <mo>⁡<!-- ⁡ --></mo>         <mi>p</mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\vartheta (x)=\\\\sum _{p\\\\leq x}\\\\ln p}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/807b37b3a5a8364c26a4e617a0db8b9feb9f3a1a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:14.849ex; height:5.843ex;" alt="{\\\\displaystyle \\\\vartheta (x)=\\\\sum _{p\\\\leq x}\\\\ln p}"></span></dd></dl> où la somme est définie sur les nombres premiers <span class="texhtml mvar" style="font-style:italic;">p</span> inférieurs ou égaux à <span class="texhtml mvar" style="font-style:italic;">x</span>. La <b>seconde fonction de Tchebychev</b> <span class="texhtml"><i>ψ</i>(<i>x</i>)</span> est définie de façon similaire, la somme s'étendant aux puissances premières inférieures à  <span class="texhtml mvar" style="font-style:italic;">x</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 \\\\psi (x)=\\\\sum _{p^{k}\\\\leq x}\\\\ln p=\\\\sum _{n\\\\leq x}\\\\Lambda (n)=\\\\sum _{p\\\\leq x}\\\\left\\\\lfloor \\\\ln _{p}x\\
ight\\
floor \\\\ln p,}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>ψ<!-- ψ --></mi>         <mo stretchy="false">(</mo>         <mi>x</mi>         <mo stretchy="false">)</mo>         <mo>=</mo>         <munder>           <mo>∑<!-- ∑ --></mo>           <mrow class="MJX-TeXAtom-ORD">             <msup>               <mi>p</mi>               <mrow class="MJX-TeXAtom-ORD">                 <mi>k</mi>               </mrow>             </msup>             <mo>≤<!-- ≤ --></mo>             <mi>x</mi>           </mrow>         </munder>         <mi>ln</mi>         <mo>⁡<!-- ⁡ --></mo>         <mi>p</mi>         <mo>=</mo>         <munder>           <mo>∑<!-- ∑ --></mo>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>             <mo>≤<!-- ≤ --></mo>             <mi>x</mi>           </mrow>         </munder>         <mi mathvariant="normal">Λ<!-- Λ --></mi>         <mo stretchy="false">(</mo>         <mi>n</mi>         <mo stretchy="false">)</mo>         <mo>=</mo>         <munder>           <mo>∑<!-- ∑ --></mo>           <mrow class="MJX-TeXAtom-ORD">             <mi>p</mi>             <mo>≤<!-- ≤ --></mo>             <mi>x</mi>           </mrow>         </munder>         <mrow>           <mo>⌊</mo>           <mrow>             <msub>               <mi>ln</mi>               <mrow class="MJX-TeXAtom-ORD">                 <mi>p</mi>               </mrow>             </msub>             <mo>⁡<!-- ⁡ --></mo>             <mi>x</mi>           </mrow>           <mo>⌋</mo>         </mrow>         <mi>ln</mi>         <mo>⁡<!-- ⁡ --></mo>         <mi>p</mi>         <mo>,</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\psi (x)=\\\\sum _{p^{k}\\\\leq x}\\\\ln p=\\\\sum _{n\\\\leq x}\\\\Lambda (n)=\\\\sum _{p\\\\leq x}\\\\left\\\\lfloor \\\\ln _{p}x\\
ight\\
floor \\\\ln p,}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b486b2356b3b6955cf82415e5df358f331d7b0c" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:45.347ex; height:6.343ex;" alt="{\\\\displaystyle \\\\psi (x)=\\\\sum _{p^{k}\\\\leq x}\\\\ln p=\\\\sum _{n\\\\leq x}\\\\Lambda (n)=\\\\sum _{p\\\\leq x}\\\\left\\\\lfloor \\\\ln _{p}x\\
ight\\
floor \\\\ln p,}"></span></dd></dl> où <span class="texhtml">Λ</span> désigne la fonction de von Mangoldt. Les fonctions de Tchebychev, notamment la seconde <span class="texhtml"><i>ψ</i>(<i>x</i>)</span>, sont souvent utilisées dans des résultats sur les nombres premiers, car elles sont plus simples à utiliser que la fonction de compte des nombres premiers, <span class="texhtml">π(<i>x</i>)</span> (voir la formule exacte, plus bas). Les deux fonctions de Tchebychev sont asymptotiquement équivalentes à <span class="texhtml mvar" style="font-style:italic;">x</span>, un résultat similaire au théorème des nombres premiers. Les deux fonctions sont nommées d'après Pafnouti Tchebychev.  
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Fonction_de_Tchebychev">https://fr.wikipedia.org/wiki/Fonction_de_Tchebychev</a>)"""@fr, """In mathematics, the <b>Chebyshev function</b> is either a scalarising function (<b>Tchebycheff function</b>) or one of two related functions.  The <b>first Chebyshev function</b> <span class="texhtml"><i>ϑ</i>  (<i>x</i>)</span> or <span class="texhtml"><i>θ</i> (<i>x</i>)</span> is given by  <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 \\\\vartheta (x)=\\\\sum _{p\\\\leq x}\\\\log p}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>ϑ<!-- ϑ --></mi>         <mo stretchy="false">(</mo>         <mi>x</mi>         <mo stretchy="false">)</mo>         <mo>=</mo>         <munder>           <mo>∑<!-- ∑ --></mo>           <mrow class="MJX-TeXAtom-ORD">             <mi>p</mi>             <mo>≤<!-- ≤ --></mo>             <mi>x</mi>           </mrow>         </munder>         <mi>log</mi>         <mo>⁡<!-- ⁡ --></mo>         <mi>p</mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\vartheta (x)=\\\\sum _{p\\\\leq x}\\\\log p}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba38682ec80afa73c29c41a52a63ea1976ee7305" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:15.882ex; height:5.843ex;" alt="{\\\\displaystyle \\\\vartheta (x)=\\\\sum _{p\\\\leq x}\\\\log p}"></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 \\\\log }">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>log</mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\log }</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79e4debd0ab1c6ce342d0172a7643733305c37bc" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.972ex; height:2.509ex;" alt="{\\\\displaystyle \\\\log }"></span> denotes the natural logarithm, with the sum extending over all prime numbers <span class="texhtml mvar" style="font-style:italic;">p</span> that are less than or equal to <span class="texhtml mvar" style="font-style:italic;">x</span>. The <b>second Chebyshev function</b> <span class="texhtml"><i>ψ</i> (<i>x</i>)</span> is defined similarly, with the sum extending over all prime powers not exceeding <span class="texhtml mvar" style="font-style:italic;">x</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 \\\\psi (x)=\\\\sum _{k\\\\in \\\\mathbb {N} }\\\\sum _{p^{k}\\\\leq x}\\\\log p=\\\\sum _{n\\\\leq x}\\\\Lambda (n)=\\\\sum _{p\\\\leq x}\\\\left\\\\lfloor \\\\log _{p}x\\
ight\\
floor \\\\log p,}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>ψ<!-- ψ --></mi>         <mo stretchy="false">(</mo>         <mi>x</mi>         <mo stretchy="false">)</mo>         <mo>=</mo>         <munder>           <mo>∑<!-- ∑ --></mo>           <mrow class="MJX-TeXAtom-ORD">             <mi>k</mi>             <mo>∈<!-- ∈ --></mo>             <mrow class="MJX-TeXAtom-ORD">               <mi mathvariant="double-struck">N</mi>             </mrow>           </mrow>         </munder>         <munder>           <mo>∑<!-- ∑ --></mo>           <mrow class="MJX-TeXAtom-ORD">             <msup>               <mi>p</mi>               <mrow class="MJX-TeXAtom-ORD">                 <mi>k</mi>               </mrow>             </msup>             <mo>≤<!-- ≤ --></mo>             <mi>x</mi>           </mrow>         </munder>         <mi>log</mi>         <mo>⁡<!-- ⁡ --></mo>         <mi>p</mi>         <mo>=</mo>         <munder>           <mo>∑<!-- ∑ --></mo>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>             <mo>≤<!-- ≤ --></mo>             <mi>x</mi>           </mrow>         </munder>         <mi mathvariant="normal">Λ<!-- Λ --></mi>         <mo stretchy="false">(</mo>         <mi>n</mi>         <mo stretchy="false">)</mo>         <mo>=</mo>         <munder>           <mo>∑<!-- ∑ --></mo>           <mrow class="MJX-TeXAtom-ORD">             <mi>p</mi>             <mo>≤<!-- ≤ --></mo>             <mi>x</mi>           </mrow>         </munder>         <mrow>           <mo>⌊</mo>           <mrow>             <msub>               <mi>log</mi>               <mrow class="MJX-TeXAtom-ORD">                 <mi>p</mi>               </mrow>             </msub>             <mo>⁡<!-- ⁡ --></mo>             <mi>x</mi>           </mrow>           <mo>⌋</mo>         </mrow>         <mi>log</mi>         <mo>⁡<!-- ⁡ --></mo>         <mi>p</mi>         <mo>,</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\psi (x)=\\\\sum _{k\\\\in \\\\mathbb {N} }\\\\sum _{p^{k}\\\\leq x}\\\\log p=\\\\sum _{n\\\\leq x}\\\\Lambda (n)=\\\\sum _{p\\\\leq x}\\\\left\\\\lfloor \\\\log _{p}x\\
ight\\
floor \\\\log p,}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9616f859e7f64da9044ce816f81c71268c7df41" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:52.316ex; height:6.343ex;" alt="{\\\\displaystyle \\\\psi (x)=\\\\sum _{k\\\\in \\\\mathbb {N} }\\\\sum _{p^{k}\\\\leq x}\\\\log p=\\\\sum _{n\\\\leq x}\\\\Lambda (n)=\\\\sum _{p\\\\leq x}\\\\left\\\\lfloor \\\\log _{p}x\\
ight\\
floor \\\\log p,}"></span></dd></dl> where <span class="texhtml">Λ</span> is the von Mangoldt function. The Chebyshev functions, especially the second one <span class="texhtml"><i>ψ</i> (<i>x</i>)</span>, are often used in proofs related to prime numbers, because it is typically simpler to work with them than with the prime-counting function, <span class="texhtml"><i>π</i> (<i>x</i>)</span> (see the exact formula below.) Both Chebyshev functions are asymptotic to <span class="texhtml mvar" style="font-style:italic;">x</span>, a statement equivalent to the prime number theorem.  
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Chebyshev_function">https://en.wikipedia.org/wiki/Chebyshev_function</a>)"""@en ;
  a skos:Concept ;
  skos:prefLabel "Chebyshev function"@en, "fonction de Tchebychev"@fr ;
  skos:inScheme psr: ;
  skos:broader psr:-MWBTPGLM-B .