Riemann-Siegel function theta: Series representations (formula 10.03.06.0032)

RiemannSiegelTheta

$Mathematica Notation$

Zeta Functions and Polylogarithms

RiemannSiegelTheta[z]

Series representations

Generalized power series

Expansions at z==0

http://functions.wolfram.com/10.03.06.0032.01

Input Form

RiemannSiegelTheta[z] \[Proportional] (-(1/4)) (2 EulerGamma + Pi + 2 Log[8 Pi]) z (1 + O[z^2]) /; (z -> 0)

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["RiemannSiegelTheta", "[", "z", "]"]], "\[Proportional]", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "EulerGamma"]], "+", "\[Pi]", "+", RowBox[List["2", " ", RowBox[List["Log", "[", RowBox[List["8", " ", "\[Pi]"]], "]"]]]]]], ")"]], "z", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", SuperscriptBox["z", "2"], "]"]]]], ")"]]]]]], "/;", RowBox[List["(", RowBox[List["z", "\[Rule]", "0"]], ")"]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <semantics> <mi> ϑ </mi> <annotation encoding='Mathematica'> TagBox["\[CurlyTheta]", RiemannSiegelTheta] </annotation> </semantics> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ∝ </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 8 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mi> π </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <semantics> <mi> ℽ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] </annotation> </semantics> </mrow> </mrow> <mn> 4 </mn> </mfrac> </mrow> <mo> ⁢ </mo> <mi> z </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> O </mi> <mo> ⁡ </mo> <mo> ( </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> RiemannSiegelTheta </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ln /> <apply> <times /> <cn type='integer'> 8 </cn> <pi /> </apply> </apply> </apply> <pi /> <apply> <times /> <cn type='integer'> 2 </cn> <eulergamma /> </apply> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> z </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <ci> O </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <ci> Rule </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["RiemannSiegelTheta", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "EulerGamma"]], "+", "\[Pi]", "+", RowBox[List["2", " ", RowBox[List["Log", "[", RowBox[List["8", " ", "\[Pi]"]], "]"]]]]]], ")"]], " ", "z", " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox[RowBox[List["O", "[", "z", "]"]], "2"]]], ")"]]]], "/;", RowBox[List["(", RowBox[List["z", "\[Rule]", "0"]], ")"]]]]]]]]

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

2007-05-02