RamificationIndex[ZetaClassical[s, a], a, -n] == q /; Element[n, Integers] && n >= 0 && s == p/q && Element[p, Integers] && Element[q - 1, Integers] && q - 1 > 0 && GCD[p, q] == 1

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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msub> <mi> &#8475; </mi> <mi> a </mi> </msub> <mo> ( </mo> <mrow> <mrow> <mover> <mi> &#950; </mi> <mo> ^ </mo> </mover> <mo> ( </mo> <mrow> <semantics> <mi> s </mi> <annotation encoding='Mathematica'> TagBox[&quot;s&quot;, Rule[Editable, True]] </annotation> </semantics> <mo> , </mo> <semantics> <mi> a </mi> <annotation encoding='Mathematica'> TagBox[&quot;a&quot;, Rule[Editable, True]] </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mi> n </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mi> q </mi> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mi> s </mi> <mo> &#10869; </mo> <mfrac> <mi> p </mi> <mi> q </mi> </mfrac> </mrow> <mo> &#8743; </mo> <mrow> <mi> p </mi> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> q </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> gcd </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> p </mi> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <apply> <ci> Subscript </ci> <ci> &#8475; </ci> <ci> a </ci> </apply> <apply> <apply> <ci> OverHat </ci> <ci> &#950; </ci> </apply> <apply> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> <ci> s </ci> </apply> <apply> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> <ci> a </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <ci> q </ci> </apply> <apply> <and /> <apply> <in /> <ci> n </ci> <ci> &#8469; </ci> </apply> <apply> <eq /> <ci> s </ci> <apply> <times /> <ci> p </ci> <apply> <power /> <ci> q </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <in /> <ci> p </ci> <integers /> </apply> <apply> <in /> <apply> <plus /> <ci> q </ci> <cn type='integer'> -1 </cn> </apply> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> <apply> <eq /> <apply> <gcd /> <ci> p </ci> <ci> q </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2001-10-29