Product[1 + Subscript[p, k]^(-s), {k, 1, Infinity}] == Zeta[s]/Zeta[2 s] /; Re[s] > 1 && Subscript[p, k] == Prime[k]

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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> <munderover> <mo> &#8719; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <semantics> <mi> &#8734; </mi> <annotation-xml encoding='MathML-Content'> <infinity /> </annotation-xml> </semantics> </munderover> <mrow> <mo> ( </mo> <mrow> <msubsup> <mi> p </mi> <mi> k </mi> <mrow> <mo> - </mo> <mi> s </mi> </mrow> </msubsup> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </mo> <mfrac> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> s </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, TagBox[&quot;s&quot;, Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> s </mi> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, TagBox[RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;s&quot;]], Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> </mfrac> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> s </mi> <mo> ) </mo> </mrow> <mo> &gt; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> p </mi> <mi> k </mi> </msub> <mo> &#10869; </mo> <mrow> <mi> prime </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <product /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <apply> <ci> DirectedInfinity </ci> <cn type='integer'> 1 </cn> </apply> </uplimit> <apply> <plus /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <apply> <ci> Zeta </ci> <ci> s </ci> </apply> <apply> <power /> <apply> <ci> Zeta </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> s </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <and /> <apply> <gt /> <apply> <real /> <ci> s </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> k </ci> </apply> <apply> <ci> prime </ci> <ci> k </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2007-05-02