|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
http://functions.wolfram.com/10.03.06.0020.01
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
RiemannSiegelTheta[z] ==
(1/8) (-4 z (1 + Log[4 Pi]) + (-I + 2 z) Log[1 - 2 I z] +
(I + 2 z) Log[1 + 2 I z]) -
(I/4) Sum[((k - 1) (Zeta[k, 5/4 + (I z)/2] - Zeta[k, 5/4 - (I z)/2]))/
(k (k + 1)), {k, 2, Infinity}]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Cell[BoxData[RowBox[List[RowBox[List["RiemannSiegelTheta", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "z", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["Log", "[", RowBox[List["4", " ", "\[Pi]"]], "]"]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], "+", RowBox[List["2", " ", "z"]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List["1", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", "+", RowBox[List["2", " ", "z"]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List["1", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]]]], "]"]]]]]], ")"]]]], "-", RowBox[List[FractionBox["\[ImaginaryI]", "4"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "2"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List["k", "-", "1"]], ")"]], RowBox[List["(", " ", RowBox[List[RowBox[List["Zeta", "[", RowBox[List["k", ",", RowBox[List[FractionBox["5", "4"], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], "2"]]]]], "]"]], "-", " ", RowBox[List["Zeta", "[", RowBox[List["k", ",", RowBox[List[FractionBox["5", "4"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], "2"]]]]], "]"]]]], ")"]]]], RowBox[List["k", " ", RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]]]]]]]]]]]]]]]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <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> <mfrac> <mn> 1 </mn> <mn> 8 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 4 </mn> </mrow> <mo> ⁢ </mo> <mi> z </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mi> ⅈ </mi> </mrow> <mo> ) </mo> </mrow> <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> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ) </mo> </mrow> <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> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mi> ⅈ </mi> <mn> 4 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 2 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <mfrac> <mrow> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mrow> <mi> k </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> , </mo> <mrow> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> + </mo> <mfrac> <mn> 5 </mn> <mn> 4 </mn> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", RowBox[List[TagBox["k", Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], "2"], "+", FractionBox["5", "4"]]], Rule[Editable, True]]]], ")"]], InterpretTemplate[Function[List[$CellContext`e1, $CellContext`e2], Zeta[$CellContext`e1, $CellContext`e2]]]] </annotation> </semantics> <mo> - </mo> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> , </mo> <mrow> <mfrac> <mn> 5 </mn> <mn> 4 </mn> </mfrac> <mo> - </mo> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", RowBox[List[TagBox["k", Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["5", "4"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], "2"]]], Rule[Editable, True]]]], ")"]], InterpretTemplate[Function[List[$CellContext`e1, $CellContext`e2], Zeta[$CellContext`e1, $CellContext`e2]]]] </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> RiemannSiegelTheta </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='rational'> 1 <sep /> 8 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -4 </cn> <ci> z </ci> <apply> <plus /> <apply> <ln /> <apply> <times /> <cn type='integer'> 4 </cn> <pi /> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <imaginaryi /> </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 /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </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> <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> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 2 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <plus /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <times /> <ci> k </ci> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <ci> Zeta </ci> <ci> k </ci> <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'> 5 <sep /> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Zeta </ci> <ci> k </ci> <apply> <plus /> <cn type='rational'> 5 <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> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
|
|
|
|
|
|
|
|
|
|
| |
|
|
|
|
| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["RiemannSiegelTheta", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "z", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["Log", "[", RowBox[List["4", " ", "\[Pi]"]], "]"]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], "+", RowBox[List["2", " ", "z"]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List["1", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", "+", RowBox[List["2", " ", "z"]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List["1", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]]]], "]"]]]]]], ")"]]]], "-", RowBox[List[FractionBox["1", "4"], " ", "\[ImaginaryI]", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "2"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List["k", "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Zeta", "[", RowBox[List["k", ",", RowBox[List[FractionBox["5", "4"], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], "2"]]]]], "]"]], "-", RowBox[List["Zeta", "[", RowBox[List["k", ",", RowBox[List[FractionBox["5", "4"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], "2"]]]]], "]"]]]], ")"]]]], RowBox[List["k", " ", RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]]]]]]]]]]]]]]] |
|
|
|
|
|
|
|
|
|
|
|
Date Added to functions.wolfram.com (modification date)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
){kind=small}
