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variants of this functions
Zeta






Mathematica Notation

Traditional Notation









Zeta Functions and Polylogarithms > Zeta[s] > Zeros > Sums over zeros





http://functions.wolfram.com/10.01.30.0005.01









  


  










Input Form





Subscript[Ζ,5]5*EulerGamma^3*StieltjesGamma[1] + (5*StieltjesGamma[2]*StieltjesGamma[1])/2 + EulerGamma^5 + (5*EulerGamma^2*StieltjesGamma[2])/2 + (5/6)*EulerGamma*(6*StieltjesGamma[1]^2 + StieltjesGamma[3]) + (5*StieltjesGamma[4])/24 - (31*Zeta[5])/32 + 1/; Subscript[Ζ,5]Limit[Sum[1/Subscript[ρ,k]^5,Abs[Subscript[ρ,k]<T]], TInfinity]&&Zeta[Subscript[ρ,k]]0&& Re[Subscript[ρ,k]]≠0










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[SubscriptBox["\[CapitalZeta]", "5"], "\[Equal]", RowBox[List["1", "+", SuperscriptBox["EulerGamma", "5"], "+", RowBox[List["5", " ", SuperscriptBox["EulerGamma", "3"], " ", RowBox[List["StieltjesGamma", "[", "1", "]"]]]], "+", RowBox[List[FractionBox["5", "2"], " ", SuperscriptBox["EulerGamma", "2"], " ", RowBox[List["StieltjesGamma", "[", "2", "]"]]]], "+", RowBox[List[FractionBox["5", "2"], " ", RowBox[List["StieltjesGamma", "[", "1", "]"]], " ", RowBox[List["StieltjesGamma", "[", "2", "]"]]]], "+", RowBox[List[FractionBox["5", "6"], " ", "EulerGamma", " ", RowBox[List["(", RowBox[List[RowBox[List["6", " ", SuperscriptBox[RowBox[List["StieltjesGamma", "[", "1", "]"]], "2"]]], "+", RowBox[List["StieltjesGamma", "[", "3", "]"]]]], ")"]]]], "+", FractionBox[RowBox[List["5", " ", RowBox[List["StieltjesGamma", "[", "4", "]"]]]], "24"], "-", FractionBox[RowBox[List["31", " ", RowBox[List["Zeta", "[", "5", "]"]]]], "32"]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["\[CapitalZeta]", "5"], "\[Equal]", RowBox[List["Limit", "[", RowBox[List[RowBox[List[UnderscriptBox["\[Sum]", RowBox[List[RowBox[List["\[LeftBracketingBar]", SubscriptBox["\[Rho]", "k"], "\[RightBracketingBar]"]], "<", "T"]]], FractionBox["1", SubsuperscriptBox["\[Rho]", "k", "5"]]]], ",", RowBox[List["T", "\[Rule]", "\[Infinity]"]]]], "]"]]]], "\[And]", RowBox[List[RowBox[List["Zeta", "[", SubscriptBox["\[Rho]", "k"], "]"]], "\[Equal]", "0"]], "\[And]", RowBox[List[RowBox[List["Re", "[", SubscriptBox["\[Rho]", "k"], "]"]], "\[NotEqual]", "0"]]]]]]]]










MathML Form







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</ms> <ci> StieltjesGamma </ci> </apply> <ms> 1 </ms> <ms> 2 </ms> </apply> </list> </apply> <ms> + </ms> <apply> <ci> SubscriptBox </ci> <apply> <ci> TagBox </ci> <ms> &#947; </ms> <ci> StieltjesGamma </ci> </apply> <ms> 3 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> + </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> 5 </ms> <apply> <ci> SubscriptBox </ci> <apply> <ci> TagBox </ci> <ms> &#947; </ms> <ci> StieltjesGamma </ci> </apply> <ms> 4 </ms> </apply> </list> </apply> <ms> 24 </ms> </apply> <ms> - </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> 31 </ms> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> &#950; </ms> <ms> ( </ms> <apply> <ci> TagBox </ci> <ms> 5 </ms> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> InterpretTemplate </ci> <apply> <ci> Function </ci> <ci> BoxForm`e$ </ci> <apply> <ci> Zeta </ci> <ci> BoxForm`e$ </ci> </apply> </apply> </apply> </apply> </list> </apply> <ms> 32 </ms> </apply> <ms> + </ms> <ms> 1 </ms> </list> </apply> </list> </apply> <ms> /; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> &#918; </ms> <ms> 5 </ms> </apply> <ms> &#10869; </ms> <apply> <ci> RowBox </ci> <list> <ms> Limit </ms> <ms> [ </ms> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderscriptBox </ci> <apply> <ci> ErrorBox </ci> <ms> &#8721; </ms> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> &#62979; </ms> <apply> <ci> SubscriptBox </ci> <ms> &#961; </ms> <ms> k </ms> </apply> <ms> &#62980; </ms> </list> </apply> <ms> = </ms> <ms> 1 </ms> </list> </apply> </apply> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <apply> <ci> SubsuperscriptBox </ci> <ms> &#961; </ms> <ms> k </ms> <ms> 5 </ms> </apply> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <ms> T </ms> <ms> -&gt; </ms> <ms> &#8734; </ms> </list> </apply> <ms> ] </ms> </list> </apply> </list> </apply> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> &#950; </ms> <ms> ( </ms> <apply> <ci> TagBox </ci> <apply> <ci> SubscriptBox </ci> <ms> &#961; </ms> <ms> k </ms> </apply> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> InterpretTemplate </ci> <apply> <ci> Function </ci> <ci> BoxForm`e$ </ci> <apply> <ci> Zeta </ci> <ci> BoxForm`e$ </ci> </apply> </apply> </apply> </apply> <ms> &#10869; </ms> <ms> 0 </ms> </list> </apply> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> Re </ms> <ms> ( </ms> <apply> <ci> SubscriptBox </ci> <ms> &#961; </ms> <ms> k </ms> </apply> <ms> ) </ms> </list> </apply> <ms> &#8800; </ms> <ms> 0 </ms> </list> </apply> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>










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





2003-08-21