@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:-VHDD6KJX-8
  skos:prefLabel "analytic number theory"@en, "théorie analytique des nombres"@fr ;
  a skos:Concept ;
  skos:narrower psr:-NXK1VBV2-9 .

psr: a skos:ConceptScheme .
psr:-NXK1VBV2-9
  skos:broader psr:-VHDD6KJX-8, psr:-FH1H1FB9-1 ;
  a skos:Concept ;
  skos:prefLabel "fonction de Buchstab"@fr, "Buchstab function"@en ;
  skos:definition """The <b>Buchstab function</b> (or <b>Buchstab's function</b>) is the unique continuous 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 \\\\omega :\\\\mathbb {R} _{\\\\geq 1}\\
ightarrow \\\\mathbb {R} _{>0}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>ω<!-- ω --></mi>
<br/>        <mo>:</mo>
<br/>        <msub>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi mathvariant="double-struck">R</mi>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mo>≥<!-- ≥ --></mo>
<br/>            <mn>1</mn>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo stretchy="false">→<!-- → --></mo>
<br/>        <msub>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi mathvariant="double-struck">R</mi>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mo>&gt;</mo>
<br/>            <mn>0</mn>
<br/>          </mrow>
<br/>        </msub>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\omega :\\\\mathbb {R} _{\\\\geq 1}\\
ightarrow \\\\mathbb {R} _{&gt;0}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/213ddd397a5e9b73567e7902d166de4e34f0c12a" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:15.019ex; height:2.676ex;" alt="\\\\omega :\\\\mathbb{R} _{{\\\\geq 1}}\\
ightarrow \\\\mathbb{R} _{{>0}}"></span> defined by the delay differential equation
<br/>
<br/><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 \\\\omega (u)={\\rac {1}{u}},\\\\qquad \\\\qquad \\\\qquad 1\\\\leq u\\\\leq 2,}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>ω<!-- ω --></mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>u</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mn>1</mn>
<br/>            <mi>u</mi>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mo>,</mo>
<br/>        <mspace width="2em"></mspace>
<br/>        <mspace width="2em"></mspace>
<br/>        <mspace width="2em"></mspace>
<br/>        <mn>1</mn>
<br/>        <mo>≤<!-- ≤ --></mo>
<br/>        <mi>u</mi>
<br/>        <mo>≤<!-- ≤ --></mo>
<br/>        <mn>2</mn>
<br/>        <mo>,</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\omega (u)={\\rac {1}{u}},\\\\qquad \\\\qquad \\\\qquad 1\\\\leq u\\\\leq 2,}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e70ac57383ab24d0f014c52bafe5d4d5565c4b6" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.838ex; width:35.317ex; height:5.176ex;" alt="\\\\omega (u)={\\rac  1u},\\\\qquad \\\\qquad \\\\qquad 1\\\\leq u\\\\leq 2,"></span></dd>
<br/><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 {\\rac {d}{du}}(u\\\\omega (u))=\\\\omega (u-1),\\\\qquad u\\\\geq 2.}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mi>d</mi>
<br/>            <mrow>
<br/>              <mi>d</mi>
<br/>              <mi>u</mi>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>u</mi>
<br/>        <mi>ω<!-- ω --></mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>u</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <mi>ω<!-- ω --></mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>u</mi>
<br/>        <mo>−<!-- − --></mo>
<br/>        <mn>1</mn>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>,</mo>
<br/>        <mspace width="2em"></mspace>
<br/>        <mi>u</mi>
<br/>        <mo>≥<!-- ≥ --></mo>
<br/>        <mn>2.</mn>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle {\\rac {d}{du}}(u\\\\omega (u))=\\\\omega (u-1),\\\\qquad u\\\\geq 2.}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2dd399ef46aa7d11ef0baf2decdea64a9ad578a8" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.005ex; width:34.708ex; height:5.509ex;" alt="{{\\rac  {d}{du}}}(u\\\\omega (u))=\\\\omega (u-1),\\\\qquad u\\\\geq 2."></span></dd></dl>
<br/>In the second equation, the derivative at <i>u</i>&nbsp;=&nbsp;2 should be taken as <i>u</i> approaches 2 from the right. It is named after Alexander Buchstab, who wrote about it in 1937.
<br/> 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Buchstab_function">https://en.wikipedia.org/wiki/Buchstab_function</a>)"""@en ;
  dc:modified "2023-08-17"^^xsd:date ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Buchstab_function> ;
  dc:created "2023-08-17"^^xsd:date ;
  skos:inScheme psr: .

psr:-FH1H1FB9-1
  skos:prefLabel "special function"@en, "fonction spéciale"@fr ;
  a skos:Concept ;
  skos:narrower psr:-NXK1VBV2-9 .