RamanujanTauL[z] == Product[1/(1 - RamanujanTau[Subscript[p, j]]/ Subscript[p, j]^z + Subscript[p, j]^(11 - 2 z)), {j, 1, Infinity}] /; Element[Subscript[p, j], Primes] && Re[z] > 13/2

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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> <mi> &#964;L </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mfrac> <mn> 1 </mn> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mrow> <semantics> <mi> &#964; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Tau]&quot;, RamanujanTau] </annotation> </semantics> <mo> ( </mo> <msub> <mi> p </mi> <mi> j </mi> </msub> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msubsup> <mi> p </mi> <mi> j </mi> <mrow> <mo> - </mo> <mi> z </mi> </mrow> </msubsup> </mrow> <mo> + </mo> <msubsup> <mi> p </mi> <mi> j </mi> <mrow> <mn> 11 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> </msubsup> </mrow> </mfrac> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <msub> <mi> p </mi> <mi> j </mi> </msub> <mo> &#8712; </mo> <semantics> <mi> &#8473; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalP]&quot;, Function[List[], Primes]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &gt; </mo> <mfrac> <mn> 13 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> &#964;L </ci> <ci> z </ci> </apply> <apply> <product /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> RamanujanTau </ci> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> j </ci> </apply> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> j </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> j </ci> </apply> <apply> <plus /> <cn type='integer'> 11 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> j </ci> </apply> <primes /> </apply> <apply> <gt /> <apply> <real /> <ci> z </ci> </apply> <cn type='rational'> 13 <sep /> 2 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2007-05-02