Re[RiemannSiegelTheta[x + I y]] == (1/2) (ArcTan[1 + 2 y, -2 x] - ArcTan[1 - 2 y, 2 x] - x (EulerGamma + Log[Pi]) + Sum[x/k - ArcTan[(1 + 4 k - 2 y)/k, (2 x)/k] + ArcTan[(1 + 4 k + 2 y)/k, -((2 x)/k)], {k, 1, Infinity}])

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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> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <semantics> <mi> &#977; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[CurlyTheta]&quot;, RiemannSiegelTheta] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> x </mi> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> y </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> y </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mi> x </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> y </mi> </mrow> </mrow> <mo> , </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> x </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> x </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#960; </mi> <mo> ) </mo> </mrow> <mo> + </mo> <semantics> <mi> &#8509; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubledGamma]&quot;, Function[EulerGamma]] </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mo> ( </mo> <mrow> <mfrac> <mi> x </mi> <mi> k </mi> </mfrac> <mo> + </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mfrac> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> y </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mi> k </mi> </mfrac> <mo> , </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> x </mi> </mrow> <mi> k </mi> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mfrac> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> y </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mi> k </mi> </mfrac> <mo> , </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> x </mi> </mrow> <mi> k </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <real /> <apply> <ci> RiemannSiegelTheta </ci> <apply> <plus /> <ci> x </ci> <apply> <times /> <imaginaryi /> <ci> y </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <arctan /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> y </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -2 </cn> <ci> x </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <arctan /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> y </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> x </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> x </ci> <apply> <plus /> <apply> <ln /> <pi /> </apply> <eulergamma /> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <plus /> <apply> <times /> <ci> x </ci> <apply> <power /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <arctan /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> y </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> x </ci> <apply> <power /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <arctan /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> y </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> x </ci> <apply> <power /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2001-10-29