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variants of this functions
Zeta






Mathematica Notation

Traditional Notation









Zeta Functions and Polylogarithms > Zeta[s] > Series representations > Generalized power series > Expansions at s==1





http://functions.wolfram.com/10.01.06.0025.01









  


  










Input Form





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]))










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["Zeta", "'"]], "[", "s", "]"]], RowBox[List["Zeta", "[", "s", "]"]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["1", RowBox[List["s", "-", "1"]]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[SubscriptBox["\[Eta]", "k"], SuperscriptBox[RowBox[List["(", RowBox[List["s", "-", "1"]], ")"]], "k"]]]]]]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List[SubscriptBox["\[Eta]", "k"], "\[Equal]", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], RowBox[List["k", "!"]]], RowBox[List["Limit", "[", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "x"], RowBox[List[FractionBox[RowBox[List["\[CapitalLambda]", "[", "j", "]"]], "j"], SuperscriptBox[RowBox[List["Log", "[", "j", "]"]], "k"]]]]], ")"]], "-", FractionBox[SuperscriptBox[RowBox[List["Log", "[", "x", "]"]], RowBox[List["k", "+", "1"]]], RowBox[List["k", "+", "1"]]]]], ",", RowBox[List["x", "\[Rule]", "\[Infinity]"]]]], "]"]]]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["\[CapitalLambda]", "[", SuperscriptBox["p", "e"], "]"]], "\[Equal]", "1"]], "/;", RowBox[List[RowBox[List["p", "\[Element]", "Primes"]], "\[And]", RowBox[List["e", "\[Element]", "Integers"]], "\[And]", RowBox[List["e", "\[GreaterEqual]", "1"]]]]]], ")"]], "\[And]", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["\[CapitalLambda]", "[", "a", "]"]], "\[Equal]", "0"]], "/;", RowBox[List[RowBox[List["Not", "[", RowBox[List["a", "\[Element]", "Primes"]], "]"]], "||", RowBox[List[SubscriptBox["\[Exists]", RowBox[List["e", ",", RowBox[List[RowBox[List["e", "\[Element]", "Integers"]], "\[And]", RowBox[List["e", "\[GreaterEqual]", "1"]]]]]]], RowBox[List["a", "\[Equal]", SuperscriptBox["p", "e"]]]]]]]]], ")"]]]], ")"]]]], ")"]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mfrac> <mrow> <msup> <mi> &#950; </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mi> s </mi> <mo> ) </mo> </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;, Zeta, Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> </mfrac> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mrow> <mi> s </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> </mrow> <mo> - </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <msub> <mi> &#951; </mi> <mi> k </mi> </msub> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> s </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> </mrow> </mrow> </mrow> </mrow> <mo> /; </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>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", FractionBox[RowBox[List[SuperscriptBox["Zeta", "\[Prime]", Rule[MultilineFunction, None]], "[", "s_", "]"]], RowBox[List["Zeta", "[", "s_", "]"]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", RowBox[List["s", "-", "1"]]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[SubscriptBox["\[Eta]", "k"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["s", "-", "1"]], ")"]], "k"]]]]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List[SubscriptBox["\[Eta]", "k"], "\[Equal]", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Limit", "[", RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "x"], FractionBox[RowBox[List[RowBox[List["\[CapitalLambda]", "[", "j", "]"]], " ", SuperscriptBox[RowBox[List["Log", "[", "j", "]"]], "k"]]], "j"]]], "-", FractionBox[SuperscriptBox[RowBox[List["Log", "[", "x", "]"]], RowBox[List["k", "+", "1"]]], RowBox[List["k", "+", "1"]]]]], ",", RowBox[List["x", "\[Rule]", "\[Infinity]"]]]], "]"]]]], RowBox[List["k", "!"]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["\[CapitalLambda]", "[", SuperscriptBox["p", "e"], "]"]], "\[Equal]", "1"]], "/;", RowBox[List[RowBox[List["p", "\[Element]", "Primes"]], "&&", RowBox[List["e", "\[Element]", "Integers"]], "&&", RowBox[List["e", "\[GreaterEqual]", "1"]]]]]], ")"]], "&&", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["\[CapitalLambda]", "[", "a", "]"]], "\[Equal]", "0"]], "/;", RowBox[List[RowBox[List["!", RowBox[List["a", "\[Element]", "Primes"]]]], "||", RowBox[List[SubscriptBox["\[Exists]", RowBox[List["e", ",", RowBox[List[RowBox[List["e", "\[Element]", "Integers"]], "&&", RowBox[List["e", "\[GreaterEqual]", "1"]]]]]]], RowBox[List["a", "\[Equal]", SuperscriptBox["p", "e"]]]]]]]]], ")"]]]]]], ")"]]]]]]]]










Contributed by





Krzysztof Maslanka










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





2007-05-02





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