Riemann zeta function: Series representations (formula 10.01.06.0005)

Zeta

$Mathematica Notation$

Zeta Functions and Polylogarithms

Zeta[s]

Series representations

Exponential Fourier series

http://functions.wolfram.com/10.01.06.0005.01

Input Form

Zeta[s] == 2 (2 Pi)^(s - 1) Gamma[1 - s] (Sin[(Pi s)/2] Sum[Cos[2 Pi k]/k^(1 - s), {k, 1, Infinity}] + Cos[(Pi s)/2] Sum[Sin[2 Pi k]/k^(1 - s), {k, 1, Infinity}]) /; Re[s] < 1

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Zeta", "[", "s", "]"]], "\[Equal]", RowBox[List["2", SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "\[Pi]"]], ")"]], RowBox[List["s", "-", "1"]]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "s"]], "]"]], " ", RowBox[List["(", " ", RowBox[List[RowBox[List[RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Pi]", " ", "s"]], "2"], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List["Cos", "[", RowBox[List["2", " ", "\[Pi]", " ", "k"]], "]"]], SuperscriptBox["k", RowBox[List["1", "-", "s"]]]]]]]], "+", " ", RowBox[List[RowBox[List["Cos", "[", FractionBox[RowBox[List["\[Pi]", " ", "s"]], "2"], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List["Sin", "[", RowBox[List["2", " ", "\[Pi]", " ", "k"]], "]"]], SuperscriptBox["k", RowBox[List["1", "-", "s"]]]]]]]]]], ")"]]]]]], "/;", " ", RowBox[List[RowBox[List["Re", "[", "s", "]"]], "<", "1"]]]]]]

MathML Form

<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[List[$CellContext`e1, $CellContext`e2], Zeta[$CellContext`e1, $CellContext`e2]]]] </annotation> </semantics> <mo> ⩵ </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> s </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> s </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> s </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> ∞ </mi> </munderover> <mfrac> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> <msup> <mi> k </mi> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> s </mi> </mrow> </msup> </mfrac> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> s </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> ∞ </mi> </munderover> <mfrac> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> <msup> <mi> k </mi> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> s </mi> </mrow> </msup> </mfrac> </mrow> </mrow> </mrow> <mo> ) </mo> </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'> <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[List[$CellContext`e1, $CellContext`e2], Zeta[$CellContext`e1, $CellContext`e2]]]] </annotation> </semantics> <mo> ⩵ </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> s </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> s </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> s </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> ∞ </mi> </munderover> <mfrac> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> <msup> <mi> k </mi> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> s </mi> </mrow> </msup> </mfrac> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> s </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> ∞ </mi> </munderover> <mfrac> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> <msup> <mi> k </mi> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> s </mi> </mrow> </msup> </mfrac> </mrow> </mrow> </mrow> <mo> ) </mo> </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> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Zeta", "[", "s_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "\[Pi]"]], ")"]], RowBox[List["s", "-", "1"]]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "s"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Pi]", " ", "s"]], "2"], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List["Cos", "[", RowBox[List["2", " ", "\[Pi]", " ", "k"]], "]"]], SuperscriptBox["k", RowBox[List["1", "-", "s"]]]]]]]], "+", RowBox[List[RowBox[List["Cos", "[", FractionBox[RowBox[List["\[Pi]", " ", "s"]], "2"], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List["Sin", "[", RowBox[List["2", " ", "\[Pi]", " ", "k"]], "]"]], SuperscriptBox["k", RowBox[List["1", "-", "s"]]]]]]]]]], ")"]]]], "/;", RowBox[List[RowBox[List["Re", "[", "s", "]"]], "<", "1"]]]]]]]]

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

2001-10-29

Zeta[s,a]