@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:-QN7CRTLH-8 .

psr:-S5KNMR6X-4
  skos:prefLabel "Lambert series"@en, "série de Lambert"@fr ;
  a skos:Concept ;
  skos:related psr:-QN7CRTLH-8 .

psr:-QN7CRTLH-8
  skos:broader psr:-TT0V0XBL-P, psr:-NHFK3Q1R-H ;
  skos:inScheme psr: ;
  skos:prefLabel "Dirichlet character"@en, "caractère de Dirichlet"@fr ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Dirichlet_character>, <https://fr.wikipedia.org/wiki/Caract%C3%A8re_de_Dirichlet> ;
  skos:definition """In analytic number theory and related branches of mathematics, a complex-valued arithmetic 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 \\\\chi :\\\\mathbb {Z} \\
ightarrow \\\\mathbb {C} }">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>χ<!-- χ --></mi>         <mo>:</mo>         <mrow class="MJX-TeXAtom-ORD">           <mi mathvariant="double-struck">Z</mi>         </mrow>         <mo stretchy="false">→<!-- → --></mo>         <mrow class="MJX-TeXAtom-ORD">           <mi mathvariant="double-struck">C</mi>         </mrow>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\chi :\\\\mathbb {Z} \\
ightarrow \\\\mathbb {C} }</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a352aab2cf1c99034d523ecae438612765233bd" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.235ex; height:2.509ex;" alt="{\\\\displaystyle \\\\chi :\\\\mathbb {Z} \\
ightarrow \\\\mathbb {C} }"></span> is a <b>Dirichlet character of modulus <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 m}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>m</mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle m}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="m"></span></b> (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 m}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>m</mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle m}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="m"></span> is a positive integer) if for all integers <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 a}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>a</mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle a}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="a"></span> and <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 b}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>b</mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle b}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="b"></span>:  <ol><li><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 \\\\chi (ab)=\\\\chi (a)\\\\chi (b);}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>χ<!-- χ --></mi>         <mo stretchy="false">(</mo>         <mi>a</mi>         <mi>b</mi>         <mo stretchy="false">)</mo>         <mo>=</mo>         <mi>χ<!-- χ --></mi>         <mo stretchy="false">(</mo>         <mi>a</mi>         <mo stretchy="false">)</mo>         <mi>χ<!-- χ --></mi>         <mo stretchy="false">(</mo>         <mi>b</mi>         <mo stretchy="false">)</mo>         <mo>;</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\chi (ab)=\\\\chi (a)\\\\chi (b);}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bf323bae5c5b1fcff28209eef34d6aead697d16" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.993ex; height:2.843ex;" alt="{\\\\displaystyle \\\\chi (ab)=\\\\chi (a)\\\\chi (b);}"></span> that is, <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 \\\\chi }">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>χ<!-- χ --></mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\chi }</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/656111758322ace96d80a9371771aa6d3de25437" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.455ex; height:2.009ex;" alt="\\\\chi "></span> is completely multiplicative.</li> <li><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 \\\\chi (a){\\egin{cases}=0&amp;{\\	ext{if }}\\\\gcd(a,m)>1\\\\\\\\\\
eq 0&amp;{\\	ext{if }}\\\\gcd(a,m)=1.\\\\end{cases}}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>χ<!-- χ --></mi>         <mo stretchy="false">(</mo>         <mi>a</mi>         <mo stretchy="false">)</mo>         <mrow class="MJX-TeXAtom-ORD">           <mrow>             <mo>{</mo>             <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false">               <mtr>                 <mtd>                   <mo>=</mo>                   <mn>0</mn>                 </mtd>                 <mtd>                   <mrow class="MJX-TeXAtom-ORD">                     <mtext>if </mtext>                   </mrow>                   <mo movablelimits="true" form="prefix">gcd</mo>                   <mo stretchy="false">(</mo>                   <mi>a</mi>                   <mo>,</mo>                   <mi>m</mi>                   <mo stretchy="false">)</mo>                   <mo>&gt;</mo>                   <mn>1</mn>                 </mtd>               </mtr>               <mtr>                 <mtd>                   <mo>≠<!-- ≠ --></mo>                   <mn>0</mn>                 </mtd>                 <mtd>                   <mrow class="MJX-TeXAtom-ORD">                     <mtext>if </mtext>                   </mrow>                   <mo movablelimits="true" form="prefix">gcd</mo>                   <mo stretchy="false">(</mo>                   <mi>a</mi>                   <mo>,</mo>                   <mi>m</mi>                   <mo stretchy="false">)</mo>                   <mo>=</mo>                   <mn>1.</mn>                 </mtd>               </mtr>             </mtable>             <mo fence="true" stretchy="true" symmetric="true"></mo>           </mrow>         </mrow>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\chi (a){\\egin{cases}=0&amp;{\\	ext{if }}\\\\gcd(a,m)&gt;1\\\\\\\\\\
eq 0&amp;{\\	ext{if }}\\\\gcd(a,m)=1.\\\\end{cases}}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e5bfa6efd5ac0a888e53bd94a7672fdf9eb031b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.762ex; height:6.176ex;" alt="{\\\\displaystyle \\\\chi (a){\\egin{cases}=0&amp;{\\	ext{if }}\\\\gcd(a,m)>1\\\\\\\\\\
eq 0&amp;{\\	ext{if }}\\\\gcd(a,m)=1.\\\\end{cases}}}"></span>  (gcd is the greatest common divisor)</li> <li><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 \\\\chi (a+m)=\\\\chi (a)}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>χ<!-- χ --></mi>         <mo stretchy="false">(</mo>         <mi>a</mi>         <mo>+</mo>         <mi>m</mi>         <mo stretchy="false">)</mo>         <mo>=</mo>         <mi>χ<!-- χ --></mi>         <mo stretchy="false">(</mo>         <mi>a</mi>         <mo stretchy="false">)</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\chi (a+m)=\\\\chi (a)}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff42132912f44f346c22d1c9cb982982ac82a51e" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.968ex; height:2.843ex;" alt="{\\\\displaystyle \\\\chi (a+m)=\\\\chi (a)}"></span>; that is, <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 \\\\chi }">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>χ<!-- χ --></mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\chi }</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/656111758322ace96d80a9371771aa6d3de25437" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.455ex; height:2.009ex;" alt="\\\\chi "></span> is periodic with period <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 m}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>m</mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle m}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="m"></span>.</li></ol> The simplest possible character, called the <b>principal character</b>, usually denoted <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 \\\\chi _{0}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msub>           <mi>χ<!-- χ --></mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>0</mn>           </mrow>         </msub>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\chi _{0}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab65ab05ef061a305d53a395741e8dec6d267986" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.509ex; height:2.009ex;" alt="\\\\chi _{0}"></span>, (see Notation below) exists for all moduli:  <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 \\\\chi _{0}(a)={\\egin{cases}0&amp;{\\	ext{if }}\\\\gcd(a,m)>1\\\\\\\\1&amp;{\\	ext{if }}\\\\gcd(a,m)=1.\\\\end{cases}}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msub>           <mi>χ<!-- χ --></mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>0</mn>           </mrow>         </msub>         <mo stretchy="false">(</mo>         <mi>a</mi>         <mo stretchy="false">)</mo>         <mo>=</mo>         <mrow class="MJX-TeXAtom-ORD">           <mrow>             <mo>{</mo>             <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false">               <mtr>                 <mtd>                   <mn>0</mn>                 </mtd>                 <mtd>                   <mrow class="MJX-TeXAtom-ORD">                     <mtext>if </mtext>                   </mrow>                   <mo movablelimits="true" form="prefix">gcd</mo>                   <mo stretchy="false">(</mo>                   <mi>a</mi>                   <mo>,</mo>                   <mi>m</mi>                   <mo stretchy="false">)</mo>                   <mo>&gt;</mo>                   <mn>1</mn>                 </mtd>               </mtr>               <mtr>                 <mtd>                   <mn>1</mn>                 </mtd>                 <mtd>                   <mrow class="MJX-TeXAtom-ORD">                     <mtext>if </mtext>                   </mrow>                   <mo movablelimits="true" form="prefix">gcd</mo>                   <mo stretchy="false">(</mo>                   <mi>a</mi>                   <mo>,</mo>                   <mi>m</mi>                   <mo stretchy="false">)</mo>                   <mo>=</mo>                   <mn>1.</mn>                 </mtd>               </mtr>             </mtable>             <mo fence="true" stretchy="true" symmetric="true"></mo>           </mrow>         </mrow>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\chi _{0}(a)={\\egin{cases}0&amp;{\\	ext{if }}\\\\gcd(a,m)&gt;1\\\\\\\\1&amp;{\\	ext{if }}\\\\gcd(a,m)=1.\\\\end{cases}}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57b4eaf6b62362b1a48821e65c55f3b9e9c9ecc4" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.462ex; height:6.176ex;" alt="{\\\\displaystyle \\\\chi _{0}(a)={\\egin{cases}0&amp;{\\	ext{if }}\\\\gcd(a,m)>1\\\\\\\\1&amp;{\\	ext{if }}\\\\gcd(a,m)=1.\\\\end{cases}}}"></span></dd></dl> The German mathematician Peter Gustav Lejeune Dirichlet—for whom the character is named—introduced these functions in his 1837 paper on primes in arithmetic progressions.  
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Dirichlet_character">https://en.wikipedia.org/wiki/Dirichlet_character</a>)"""@en, """En mathématiques, et plus précisément en arithmétique modulaire, un caractère de Dirichlet est une fonction particulière sur un ensemble de classes de congruences sur les entiers et à valeurs complexes. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Caract%C3%A8re_de_Dirichlet">https://fr.wikipedia.org/wiki/Caract%C3%A8re_de_Dirichlet</a>)"""@fr ;
  dc:modified "2024-10-18"^^xsd:date ;
  skos:related psr:-N9QKXZ16-1, psr:-S5KNMR6X-4, psr:-BB5T3KXT-X, psr:-FLT2NQ1G-3 ;
  dc:created "2023-08-04"^^xsd:date ;
  a skos:Concept .

psr:-FLT2NQ1G-3
  skos:prefLabel "Jacobi sum"@en, "somme de Jacobi"@fr ;
  a skos:Concept ;
  skos:related psr:-QN7CRTLH-8 .

psr:-TT0V0XBL-P
  skos:prefLabel "entier quadratique"@fr, "quadratic integer"@en ;
  a skos:Concept ;
  skos:narrower psr:-QN7CRTLH-8 .

psr:-BB5T3KXT-X
  skos:prefLabel "Hecke character"@en, "caractère de Hecke"@fr ;
  a skos:Concept ;
  skos:related psr:-QN7CRTLH-8 .

psr:-N9QKXZ16-1
  skos:prefLabel "Gauss sum"@en, "somme de Gauss"@fr ;
  a skos:Concept ;
  skos:related psr:-QN7CRTLH-8 .

psr: a skos:ConceptScheme .