Integrate[PrimePi[t]/(t (t^s - 1)), {t, 2, Infinity}] == Log[Zeta[s]]/s /; 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> <msubsup> <mo> &#8747; </mo> <mn> 2 </mn> <mi> &#8734; </mi> </msubsup> <mrow> <mfrac> <mrow> <semantics> <mi> &#960; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Pi]&quot;, PrimePi] </annotation> </semantics> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> <mrow> <mi> t </mi> <mo> &#8290; </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> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mfrac> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <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> ) </mo> </mrow> <mi> s </mi> </mfrac> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> s </mi> <mo> ) </mo> </mrow> <mo> &gt; </mo> <mn> 1 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <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> <times /> <apply> <ln /> <apply> <ci> Zeta </ci> <ci> s </ci> </apply> </apply> <apply> <power /> <ci> s </ci> <cn type='integer'> -1 </cn> </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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2001-10-29