RiemannSiegelZ[z] == E^(I RiemannSiegelTheta[z]) Zeta[I z + 1/2]

Cell[BoxData[RowBox[List[RowBox[List["RiemannSiegelZ", "[", "z", "]"]], "\[Equal]", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", RowBox[List["RiemannSiegelTheta", "[", "z", "]"]]]]], " ", RowBox[List["Zeta", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "z"]], "+", FractionBox["1", "2"]]], "]"]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <semantics> <mi> Z </mi> <annotation encoding='Mathematica'> TagBox[&quot;Z&quot;, RiemannSiegelZ] </annotation> </semantics> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#977; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[CurlyTheta]&quot;, RiemannSiegelTheta] </annotation> </semantics> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </msup> <mo> &#8290; </mo> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, TagBox[RowBox[List[RowBox[List[&quot;\[ImaginaryI]&quot;, &quot; &quot;, &quot;z&quot;]], &quot;+&quot;, FractionBox[&quot;1&quot;, &quot;2&quot;]]], Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[$CellContext`e, Zeta[$CellContext`e]]]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> RiemannSiegelZ </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <apply> <ci> RiemannSiegelTheta </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <ci> Zeta </ci> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["RiemannSiegelZ", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", RowBox[List["RiemannSiegelTheta", "[", "z", "]"]]]]], " ", RowBox[List["Zeta", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "z"]], "+", FractionBox["1", "2"]]], "]"]]]]]]]]

2001-10-29