Riemann zeta function: Series representations (formula 10.01.06.0003)

Zeta

$Mathematica Notation$

Zeta Functions and Polylogarithms

Zeta[s]

Series representations

Generalized power series

Expansions at s==-2n

http://functions.wolfram.com/10.01.06.0003.01

Input Form

Zeta[s] \[Proportional] (((-1)^n (2 n)!)/(4 (2 Pi)^(2 n))) (2 Zeta[1 + 2 n] (s + 2 n) + ((Log[4 Pi^2] - 2 PolyGamma[1 + 2 n]) Zeta[1 + 2 n] - 2 Derivative[1][Zeta][1 + 2 n] + O[(s + 2 n)^3])) /; (s -> -2 n) && Element[n, Integers] && n > 0

Standard Form

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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", Zeta, Rule[Editable, True]], ")"]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> <mo> ∝ </mo> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> <mo> ⁢ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mtext> </mtext> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> <mo> + </mo> <mi> s </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox[RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]], Zeta, Rule[Editable, True]], ")"]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> </mrow> <mo> + </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <msup> <mi> π </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox[RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]], Zeta, Rule[Editable, True]], ")"]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <msup> <mi> ζ </mi> <mo> ′ </mo> </msup> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> O </mi> <mo> ⁡ </mo> <mo> ( </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> <mo> + </mo> <mi> s </mi> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> s </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <mi> n </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ∧ </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> Zeta </ci> <ci> s </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <factorial /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <ci> s </ci> </apply> <apply> <ci> Zeta </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <ln /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> PolyGamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <ci> Zeta </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> D </ci> <apply> <ci> Zeta </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <apply> <ci> O </ci> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <ci> s </ci> </apply> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <ci> Rule </ci> <ci> s </ci> <apply> <times /> <cn type='integer'> -2 </cn> <ci> n </ci> </apply> </apply> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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

2001-10-29

Zeta[s,a]