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variants of this functions
Zeta




Mathematica Notation

Traditional Notation









Zeta Functions and Polylogarithms > Zeta[s] > Product representations





http://functions.wolfram.com/10.01.08.0002.01









  


  








Input Form





Zeta[s] == (Exp[Log[2 Pi] - EulerGamma/2 - 1]/(2 (s - 1) Gamma[s/2 - 1])) Product[(1 - s/Subscript[\[Rho], k]) E^(s/Subscript[\[Rho], k]), {k, 1, Infinity}] /; Zeta[Subscript[\[Rho], k]] == 0 && Im[Subscript[\[Rho], k]] != 0








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Zeta", "[", "s", "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["Exp", "[", RowBox[List[RowBox[List["Log", "[", RowBox[List["2", " ", "\[Pi]"]], "]"]], "-", FractionBox["EulerGamma", "2"], "-", "1"]], "]"]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["s", "-", "1"]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["s", "2"], "-", "1"]], "]"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List[RowBox[List["(", RowBox[List["1", "-", FractionBox["s", SubscriptBox["\[Rho]", "k"]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", FractionBox["s", SubscriptBox["\[Rho]", "k"]]]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Zeta", "[", " ", SubscriptBox["\[Rho]", "k"], "]"]], "\[Equal]", "0"]], "\[And]", RowBox[List[RowBox[List["Im", "[", SubscriptBox["\[Rho]", "k"], "]"]], "\[NotEqual]", "0"]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <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[$CellContext`e, Zeta[$CellContext`e]]]] </annotation> </semantics> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> s </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mi> s </mi> <mn> 2 </mn> </mfrac> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mi> exp </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mfrac> <semantics> <mi> &#8509; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubledGamma]&quot;, Function[EulerGamma]] </annotation> </semantics> <mn> 2 </mn> </mfrac> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mi> s </mi> <msub> <mi> &#961; </mi> <mi> k </mi> </msub> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mfrac> <mi> s </mi> <msub> <mi> &#961; </mi> <mi> k </mi> </msub> </mfrac> </msup> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> &#961; </mi> <mi> k </mi> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, TagBox[SubscriptBox[&quot;\[Rho]&quot;, &quot;k&quot;], Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[$CellContext`e, Zeta[$CellContext`e]]]] </annotation> </semantics> <mo> &#10869; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> &#961; </mi> <mi> k </mi> </msub> <mo> ) </mo> </mrow> <mo> &#8800; </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> Zeta </ci> <ci> s </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> s </ci> <cn type='integer'> -1 </cn> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <ci> s </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <exp /> <apply> <plus /> <apply> <ln /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <eulergamma /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <product /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> s </ci> <apply> <power /> <apply> <ci> ZetaZero </ci> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <ci> s </ci> <apply> <power /> <apply> <ci> ZetaZero </ci> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <apply> <ci> Zeta </ci> <apply> <ci> ZetaZero </ci> <ci> k </ci> </apply> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <neq /> <apply> <imaginary /> <apply> <ci> ZetaZero </ci> <ci> k </ci> </apply> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Zeta", "[", "s_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["Log", "[", RowBox[List["2", " ", "\[Pi]"]], "]"]], "-", FractionBox["EulerGamma", "2"], "-", "1"]]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List[RowBox[List["(", RowBox[List["1", "-", FractionBox["s", SubscriptBox["\[Rho]", "k"]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", FractionBox["s", SubscriptBox["\[Rho]", "k"]]]]]]]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["s", "-", "1"]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["s", "2"], "-", "1"]], "]"]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Zeta", "[", SubscriptBox["\[Rho]", "k"], "]"]], "\[Equal]", "0"]], "&&", RowBox[List[RowBox[List["Im", "[", SubscriptBox["\[Rho]", "k"], "]"]], "\[NotEqual]", "0"]]]]]]]]]]


Date Added to functions.wolfram.com (modification date)





2001-10-29