Riemann zeta function: Series representations (formula 10.01.06.0018)

Zeta

$Mathematica Notation$

Zeta Functions and Polylogarithms

Zeta[s]

Series representations

Generalized power series

Expansions at s==s₀/;s₀!=1

http://functions.wolfram.com/10.01.06.0018.01

Input Form

Zeta[s] \[Proportional] Zeta[Subscript[s, 0]] + Derivative[1][Zeta][Subscript[s, 0]] (s - Subscript[s, 0]) + (1/2) Derivative[2][Zeta][Subscript[s, 0]] (s - Subscript[s, 0])^2 + O[(s - Subscript[s, 0])^3] /; Subscript[s, 0] != 1

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Zeta", "[", "s", "]"]], "\[Proportional]", RowBox[List[RowBox[List["Zeta", "[", SubscriptBox["s", "0"], "]"]], "+", " ", RowBox[List[RowBox[List[SuperscriptBox["Zeta", "\[Prime]", Rule[MultilineFunction, None]], "[", SubscriptBox["s", "0"], "]"]], RowBox[List["(", RowBox[List["s", "-", SubscriptBox["s", "0"]]], ")"]]]], "+", RowBox[List[FractionBox["1", "2"], RowBox[List[SuperscriptBox["Zeta", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", SubscriptBox["s", "0"], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["s", "-", SubscriptBox["s", "0"]]], ")"]], "2"]]], "+", RowBox[List["O", "[", SuperscriptBox[RowBox[List["(", RowBox[List["s", "-", SubscriptBox["s", "0"]]], ")"]], "3"], "]"]]]]]], "/;", RowBox[List[SubscriptBox["s", "0"], "\[NotEqual]", "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", Zeta, Rule[Editable, True]], ")"]], InterpretTemplate[Function[List[ZetaDump`e1$, ZetaDump`e2$], Zeta[ZetaDump`e1$, ZetaDump`e2$]]]] </annotation> </semantics> <mo> ∝ </mo> <mrow> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> s </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox[SubscriptBox["s", "0"], Zeta, Rule[Editable, True]], ")"]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> <mo> + </mo> <mrow> <mrow> <msup> <mi> ζ </mi> <mo> ′ </mo> </msup> <mo> ( </mo> <msub> <mi> s </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> s </mi> <mo> - </mo> <msub> <mi> s </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <msup> <mi> ζ </mi> <mi> ′′ </mi> </msup> <mo> ( </mo> <msub> <mi> s </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> s </mi> <mo> - </mo> <msub> <mi> s </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mi> O </mi> <mo> ⁡ </mo> <mo> ( </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> s </mi> <mo> - </mo> <msub> <mi> s </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <msub> <mi> s </mi> <mn> 0 </mn> </msub> <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", Zeta, Rule[Editable, True]], ")"]], InterpretTemplate[Function[List[ZetaDump`e1$, ZetaDump`e2$], Zeta[ZetaDump`e1$, ZetaDump`e2$]]]] </annotation> </semantics> <mo> ∝ </mo> <mrow> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> s </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox[SubscriptBox["s", "0"], Zeta, Rule[Editable, True]], ")"]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> <mo> + </mo> <mrow> <mrow> <msup> <mi> ζ </mi> <mo> ′ </mo> </msup> <mo> ( </mo> <msub> <mi> s </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> s </mi> <mo> - </mo> <msub> <mi> s </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <msup> <mi> ζ </mi> <mi> ′′ </mi> </msup> <mo> ( </mo> <msub> <mi> s </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> s </mi> <mo> - </mo> <msub> <mi> s </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mi> O </mi> <mo> ⁡ </mo> <mo> ( </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> s </mi> <mo> - </mo> <msub> <mi> s </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <msub> <mi> s </mi> <mn> 0 </mn> </msub> <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[RowBox[List["Zeta", "[", SubscriptBox["ss", "0"], "]"]], "+", RowBox[List[RowBox[List[SuperscriptBox["Zeta", "\[Prime]", Rule[MultilineFunction, None]], "[", SubscriptBox["ss", "0"], "]"]], " ", RowBox[List["(", RowBox[List["s", "-", SubscriptBox["ss", "0"]]], ")"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List[SuperscriptBox["Zeta", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", SubscriptBox["ss", "0"], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["s", "-", SubscriptBox["ss", "0"]]], ")"]], "2"]]], "+", SuperscriptBox[RowBox[List["O", "[", RowBox[List["s", "-", SubscriptBox["ss", "0"]]], "]"]], "3"]]], "/;", RowBox[List[SubscriptBox["ss", "0"], "\[NotEqual]", "1"]]]]]]]]

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

2007-05-02

Zeta[s,a]