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http://functions.wolfram.com/10.01.07.0014.01
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Log[Zeta[s]] == s Integrate[PrimePi[t]/(t (t^s - 1)), {t, 2, Infinity}] /;
Re[s] > 1
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Log", "[", RowBox[List["Zeta", "[", "s", "]"]], "]"]], "\[Equal]", RowBox[List["s", RowBox[List[SubsuperscriptBox["\[Integral]", "2", "\[Infinity]"], RowBox[List[FractionBox[RowBox[List["PrimePi", "[", "t", "]"]], RowBox[List["t", " ", RowBox[List["(", RowBox[List[SuperscriptBox["t", "s"], "-", "1"]], ")"]]]]], RowBox[List["\[DifferentialD]", "t"]]]]]]]]]], "/;", RowBox[List[RowBox[List["Re", "[", "s", "]"]], ">", "1"]]]]]]
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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> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <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[$CellContext`e, Zeta[$CellContext`e]]]] </annotation> </semantics> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mi> s </mi> <mo> ⁢ </mo> <mrow> <msubsup> <mo> ∫ </mo> <mn> 2 </mn> <mi> ∞ </mi> </msubsup> <mrow> <mfrac> <mrow> <semantics> <mi> π </mi> <annotation encoding='Mathematica'> TagBox["\[Pi]", PrimePi] </annotation> </semantics> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> <mrow> <mi> t </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> t </mi> <mi> s </mi> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> t </mi> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> s </mi> <mo> ) </mo> </mrow> <mo> > </mo> <mn> 1 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ln /> <apply> <ci> Zeta </ci> <ci> s </ci> </apply> </apply> <apply> <times /> <ci> s </ci> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <cn type='integer'> 2 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <ci> PrimePi </ci> <ci> t </ci> </apply> <apply> <power /> <apply> <times /> <ci> t </ci> <apply> <plus /> <apply> <power /> <ci> t </ci> <ci> s </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <gt /> <apply> <real /> <ci> s </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Log", "[", RowBox[List["Zeta", "[", "s_", "]"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["s", " ", RowBox[List[SubsuperscriptBox["\[Integral]", "2", "\[Infinity]"], RowBox[List[FractionBox[RowBox[List["PrimePi", "[", "t", "]"]], RowBox[List["t", " ", RowBox[List["(", RowBox[List[SuperscriptBox["t", "s"], "-", "1"]], ")"]]]]], RowBox[List["\[DifferentialD]", "t"]]]]]]]], "/;", RowBox[List[RowBox[List["Re", "[", "s", "]"]], ">", "1"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
