Derivative[1][Zeta][s]/Zeta[s] == -(1/(s - 1)) - Sum[Subscript[\[Eta], k] (s - 1)^k, {k, 0, Infinity}] /; (Subscript[\[Eta], k] == ((-1)^k/k!) Limit[Sum[(\[CapitalLambda][j]/j) Log[j]^k, {j, 1, x}] - Log[x]^(k + 1)/(k + 1), x -> Infinity] /; (\[CapitalLambda][p^e] == 1 /; Element[p, Primes] && Element[e, Integers] && e >= 1) && (\[CapitalLambda][a] == 0 /; !Element[a, Primes] || Exists[e, Element[e, Integers] && e >= 1, a == p^e]))

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</mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <mi> &#951; </mi> <mi> k </mi> </msub> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mtext> </mtext> </mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <munder> <mi> lim </mi> <mrow> <mi> x </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mi> &#8734; </mi> </mrow> </munder> <mo> &#8290; </mo> <mtext> &#8201; </mtext> <mrow> <mo> ( </mo> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> x </mi> </munderover> <mfrac> <mrow> <mrow> <mi> &#923; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> j </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> log </mi> <mi> k </mi> </msup> <mo> ( </mo> <mi> j </mi> <mo> ) </mo> </mrow> </mrow> <mi> j </mi> </mfrac> </mrow> <mo> - </mo> <mfrac> <mrow> <msup> <mi> log </mi> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> &#923; </mi> <mo> &#8289; </mo> <mo> ( </mo> <msup> <mi> p </mi> <mi> e </mi> </msup> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mn> 1 </mn> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> p </mi> <mo> &#8712; </mo> <semantics> <mi> &#8473; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalP]&quot;, Function[List[], Primes]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> e </mi> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> e </mi> <mo> &#8805; </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8743; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> &#923; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mn> 0 </mn> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> &#172; </mo> <mrow> <mi> a </mi> <mo> &#8712; </mo> <semantics> <mi> &#8473; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalP]&quot;, Function[List[], Primes]] </annotation> </semantics> </mrow> </mrow> <mo> &#8744; </mo> <mrow> <msub> <mo> &#8707; </mo> <mrow> <mi> e </mi> <mo> , </mo> <mrow> <mrow> <mi> e </mi> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> e </mi> <mo> &#8805; </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> </msub> <mrow> <mi> a </mi> <mo> &#10869; </mo> <msup> <mi> p </mi> <mi> e </mi> </msup> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <times /> <apply> <partialdiff /> <bvar> <ci> s </ci> </bvar> <apply> <ci> Zeta </ci> <ci> s </ci> </apply> </apply> <apply> <power /> <apply> <ci> Zeta </ci> <ci> s </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <ci> s </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <ci> Subscript </ci> <ci> &#951; </ci> <ci> k </ci> </apply> <apply> <power /> <apply> <plus /> <ci> s </ci> <cn type='integer'> -1 </cn> </apply> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> &#951; </ci> <ci> k </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <power /> <apply> <factorial /> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <limit /> <bvar> <ci> x </ci> </bvar> <condition> <apply> <tendsto /> <ci> x </ci> <infinity /> </apply> </condition> <apply> <plus /> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> x </ci> </uplimit> <apply> <times /> <apply> <ci> &#923; </ci> <ci> j </ci> </apply> <apply> <power /> <apply> <ln /> <ci> j </ci> </apply> <ci> k </ci> </apply> <apply> <power /> <ci> j </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <ln /> <ci> x </ci> </apply> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> &#923; </ci> <apply> <power /> <ci> p </ci> <ci> e </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <and /> <apply> <in /> <ci> p </ci> <primes /> </apply> <apply> <in /> <ci> e </ci> <integers /> </apply> <apply> <geq /> <ci> e </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> &#923; </ci> <ci> a </ci> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <or /> <apply> <not /> <apply> <in /> <ci> a </ci> <primes /> </apply> </apply> <apply> <exists /> <bvar> <ci> e </ci> </bvar> <bvar> <apply> <and /> <apply> <in /> <ci> e </ci> <integers /> </apply> <apply> <geq /> <ci> e </ci> <cn type='integer'> 1 </cn> </apply> </apply> </bvar> <apply> <eq /> <ci> a </ci> <apply> <power /> <ci> p </ci> <ci> e </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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Krzysztof Maslanka

2007-05-02