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http://functions.wolfram.com/10.01.07.0011.02
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Zeta[s] == Sum[k^(-s), {k, 1, n}] + n^(1 - s)/(s - 1) -
s Integrate[(t - Floor[t])/t^(s + 1), {t, n, Infinity}] /;
Element[n, Integers] && n > 0 && Re[s] > 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Zeta", "[", "s", "]"]], "\[Equal]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], SuperscriptBox["k", RowBox[List["-", "s"]]]]], "+", FractionBox[SuperscriptBox["n", RowBox[List["1", "-", "s"]]], RowBox[List["s", "-", "1"]]], "-", RowBox[List["s", RowBox[List[SubsuperscriptBox["\[Integral]", "n", "\[Infinity]"], RowBox[List[FractionBox[RowBox[List["t", "-", RowBox[List["Floor", "[", "t", "]"]]]], SuperscriptBox["t", RowBox[List["s", "+", "1"]]]], RowBox[List["\[DifferentialD]", "t"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]], "\[And]", RowBox[List[RowBox[List["Re", "[", "s", "]"]], ">", "0"]]]]]]]]
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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> <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> <mfrac> <msup> <mi> n </mi> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> s </mi> </mrow> </msup> <mrow> <mi> s </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> - </mo> <mrow> <mi> s </mi> <mo> ⁢ </mo> <mrow> <msubsup> <mo> ∫ </mo> <mi> n </mi> <mi> ∞ </mi> </msubsup> <mrow> <mfrac> <mrow> <mi> t </mi> <mo> - </mo> <mrow> <mo> ⌊ </mo> <mi> t </mi> <mo> ⌋ </mo> </mrow> </mrow> <msup> <mi> t </mi> <mrow> <mi> s </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> t </mi> </mrow> </mrow> </mrow> </mrow> <mo> + </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <msup> <mi> k </mi> <mrow> <mo> - </mo> <mi> s </mi> </mrow> </msup> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> s </mi> <mo> ) </mo> </mrow> <mo> > </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> Zeta </ci> <ci> s </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> n </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> s </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> s </ci> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <ci> n </ci> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <plus /> <ci> t </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <floor /> <ci> t </ci> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <ci> t </ci> <apply> <plus /> <ci> s </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <power /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> <apply> <gt /> <apply> <real /> <ci> s </ci> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Zeta", "[", "s_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], SuperscriptBox["k", RowBox[List["-", "s"]]]]], "+", FractionBox[SuperscriptBox["n", RowBox[List["1", "-", "s"]]], RowBox[List["s", "-", "1"]]], "-", RowBox[List["s", " ", RowBox[List[SubsuperscriptBox["\[Integral]", "n", "\[Infinity]"], RowBox[List[FractionBox[RowBox[List["t", "-", RowBox[List["Floor", "[", "t", "]"]]]], SuperscriptBox["t", RowBox[List["s", "+", "1"]]]], RowBox[List["\[DifferentialD]", "t"]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]], "&&", RowBox[List[RowBox[List["Re", "[", "s", "]"]], ">", "0"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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