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http://functions.wolfram.com/10.03.06.0019.01
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RiemannSiegelTheta[z] == (-(Log[Pi]/2)) z - (1/2) EulerGamma z -
(I/2) Log[1 - 2 I z] + (I/2) Log[1 + 2 I z] -
(I/4) Log[(Pi (1/4 + (I z)/2))/Sin[Pi (1/4 + (I z)/2)]] +
(I/4) Log[(Pi (1/4 - (I z)/2))/Sin[Pi (1/4 - (I z)/2)]] -
(1/2) Sum[((-1)^(j + k) 2^(-2 j - 2 k) z^(1 - 2 j + 2 k) (2 k)!
Zeta[1 + 2 k])/((2 j)! (2 k - 2 j + 1)!), {k, 1, Infinity},
{j, 0, k}] /; Abs[z] < 2
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["RiemannSiegelTheta", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List["Log", "[", "\[Pi]", "]"]], "2"]]], " ", "z"]], "-", RowBox[List[FractionBox["1", "2"], " ", "EulerGamma", " ", "z"]], "-", RowBox[List[FractionBox["\[ImaginaryI]", "2"], RowBox[List["Log", "[", RowBox[List["1", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]]]], "]"]]]], "+", RowBox[List[FractionBox["\[ImaginaryI]", "2"], RowBox[List["Log", "[", RowBox[List["1", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]]]], "]"]]]], "-", RowBox[List[FractionBox["\[ImaginaryI]", "4"], " ", RowBox[List["Log", "[", FractionBox[RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[FractionBox["1", "4"], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], "2"]]], ")"]]]], RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[FractionBox["1", "4"], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], "2"]]], ")"]]]], "]"]]], "]"]]]], "+", RowBox[List[FractionBox["\[ImaginaryI]", "4"], " ", RowBox[List["Log", "[", FractionBox[RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[FractionBox["1", "4"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], "2"]]], ")"]]]], RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[FractionBox["1", "4"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], "2"]]], ")"]]]], "]"]]], "]"]]]], "-", RowBox[List[FractionBox["1", "2"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["j", "+", "k"]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "j"]], "-", RowBox[List["2", " ", "k"]]]]], " ", SuperscriptBox["z", RowBox[List["1", "-", RowBox[List["2", " ", "j"]], "+", RowBox[List["2", " ", "k"]]]]], " ", RowBox[List[RowBox[List["(", RowBox[List["2", "k"]], ")"]], "!"]], " ", RowBox[List["Zeta", "[", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]], "]"]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["2", "j"]], ")"]], "!"]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "k"]], "-", RowBox[List["2", "j"]], "+", "1"]], ")"]], "!"]]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "<", "2"]]]]]]
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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> <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> <mrow> <mo> - </mo> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> π </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mfrac> <mrow> <semantics> <mi> ℽ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubledGamma]", Function[EulerGamma]] </annotation> </semantics> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> + </mo> <mrow> <mfrac> <mi> ⅈ </mi> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mi> ⅈ </mi> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mi> ⅈ </mi> <mn> 4 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mi> ⅈ </mi> <mn> 4 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> - </mo> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> - </mo> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> k </mi> </munderover> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> j </mi> <mo> + </mo> <mi> k </mi> </mrow> </msup> <mo> ⁢ </mo> <msup> <mn> 2 </mn> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <mi> j </mi> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> </mrow> </msup> <mo> ⁢ </mo> <mtext> </mtext> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox[RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], Rule[Editable, True]], ")"]], InterpretTemplate[Function[$CellContext`e, Zeta[$CellContext`e]]]] </annotation> </semantics> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> j </mi> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> </mrow> </msup> </mrow> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> j </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> j </mi> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> </mfrac> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> z </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> < </mo> <mn> 2 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> RiemannSiegelTheta </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ln /> <pi /> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <eulergamma /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <imaginaryi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <imaginaryi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <imaginaryi /> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ln /> <apply> <times /> <pi /> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <power /> <apply> <sin /> <apply> <times /> <pi /> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 4 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <imaginaryi /> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ln /> <apply> <times /> <pi /> <apply> <plus /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <sin /> <apply> <times /> <pi /> <apply> <plus /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> k </ci> </uplimit> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> j </ci> <ci> k </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <ci> j </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> </apply> </apply> <apply> <factorial /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> <apply> <ci> Zeta </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> j </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> j </ci> </apply> </apply> <apply> <factorial /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> j </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <lt /> <apply> <abs /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["RiemannSiegelTheta", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["-", RowBox[List["Log", "[", "\[Pi]", "]"]]]], ")"]], " ", "z"]], "-", FractionBox[RowBox[List["EulerGamma", " ", "z"]], "2"], "-", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", RowBox[List["Log", "[", RowBox[List["1", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]]]], "]"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", RowBox[List["Log", "[", RowBox[List["1", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]]]], "]"]]]], "-", RowBox[List[FractionBox["1", "4"], " ", "\[ImaginaryI]", " ", RowBox[List["Log", "[", FractionBox[RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[FractionBox["1", "4"], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], "2"]]], ")"]]]], RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[FractionBox["1", "4"], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], "2"]]], ")"]]]], "]"]]], "]"]]]], "+", RowBox[List[FractionBox["1", "4"], " ", "\[ImaginaryI]", " ", RowBox[List["Log", "[", FractionBox[RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[FractionBox["1", "4"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], "2"]]], ")"]]]], RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[FractionBox["1", "4"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], "2"]]], ")"]]]], "]"]]], "]"]]]], "-", RowBox[List[FractionBox["1", "2"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["j", "+", "k"]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "j"]], "-", RowBox[List["2", " ", "k"]]]]], " ", SuperscriptBox["z", RowBox[List["1", "-", RowBox[List["2", " ", "j"]], "+", RowBox[List["2", " ", "k"]]]]], " ", RowBox[List[RowBox[List["(", RowBox[List["2", " ", "k"]], ")"]], "!"]], " ", RowBox[List["Zeta", "[", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]], "]"]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["2", " ", "j"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", RowBox[List["2", " ", "j"]], "+", "1"]], ")"]], "!"]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "<", "2"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
