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http://functions.wolfram.com/13.03.24.0005.01
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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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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], InterpretationBox["\[Infinity]", DirectedInfinity[1]]], RowBox[List["(", RowBox[List["1", "+", SubsuperscriptBox["p", "k", RowBox[List["-", "s"]]]]], ")"]]]], "\[Equal]", FractionBox[RowBox[List["Zeta", "[", "s", "]"]], RowBox[List["Zeta", "[", RowBox[List["2", " ", "s"]], "]"]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", "s", "]"]], ">", "1"]], "\[And]", RowBox[List[SubscriptBox["p", "k"], "\[Equal]", RowBox[List["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> ∏ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <semantics> <mi> ∞ </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> ⩵ </mo> <mfrac> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> s </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox["s", Rule[Editable, True]], ")"]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> s </mi> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox[RowBox[List["2", " ", "s"]], Rule[Editable, True]], ")"]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> </mfrac> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> s </mi> <mo> ) </mo> </mrow> <mo> > </mo> <mn> 1 </mn> </mrow> <mo> ∧ </mo> <mrow> <msub> <mi> p </mi> <mi> k </mi> </msub> <mo> ⩵ </mo> <mrow> <mi> prime </mi> <mo> ⁡ </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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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List["(", RowBox[List["1", "+", SubsuperscriptBox["p_", "k", RowBox[List["-", "s_"]]]]], ")"]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["Zeta", "[", "s", "]"]], RowBox[List["Zeta", "[", RowBox[List["2", " ", "s"]], "]"]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", "s", "]"]], ">", "1"]], "&&", RowBox[List[SubscriptBox["p", "k"], "\[Equal]", RowBox[List["Prime", "[", "k", "]"]]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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