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RiemannSiegelZ






Mathematica Notation

Traditional Notation









Zeta Functions and Polylogarithms > RiemannSiegelZ[z] > Series representations > Asymptotic series expansions





http://functions.wolfram.com/10.04.06.0010.01









  


  










Input Form





RiemannSiegelZ[x] \[Proportional] 2 Sum[Cos[RiemannSiegelTheta[x] - x Log[k]]/Sqrt[k], {k, 1, \[Nu]}] + (-1)^(\[Nu] - 1) (2 Pi)^(1/4) (\[CapitalOmega][p]/x^4^(-1) - (1/(x^(3/4) (48 Sqrt[2] Pi^(3/2)))) Derivative[3][\[CapitalOmega]][p] + (2 Pi (Derivative[2][\[CapitalOmega]][p]/(64 Pi^2) + Derivative[6][\[CapitalOmega]][p]/(18432 Pi^4)))/x^(5/4) - ((2 Pi)^(3/2) (Derivative[1][\[CapitalOmega]][p]/(64 Pi^2) + Derivative[5][\[CapitalOmega]][p]/(3840 Pi^4) + Derivative[9][\[CapitalOmega]][p]/(5308416 Pi^6)))/x^(7/4)) /; \[Nu] == Floor[Sqrt[x/(2 Pi)]] && p == Sqrt[x/(2 Pi)] - \[Nu] && \[CapitalOmega][p] == Cos[2 Pi (p^2 - p - 1/16)]/Cos[2 Pi p] && Element[x, Reals] && (x -> Infinity)










Standard Form





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MathML Form







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</ci> <apply> <floor /> <apply> <power /> <apply> <times /> <ci> x </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <eq /> <ci> p </ci> <apply> <plus /> <apply> <power /> <apply> <times /> <ci> x </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> </apply> <apply> <eq /> <apply> <ci> &#937; </ci> <ci> p </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <cos /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <ci> p </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <cos /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <apply> <plus /> <apply> <power /> <ci> p </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> p </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 16 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> x </ci> <reals /> </apply> <apply> <ci> Rule </ci> <ci> x </ci> <infinity /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["RiemannSiegelZ", "[", "x_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["2", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Nu]"], FractionBox[RowBox[List["Cos", "[", RowBox[List[RowBox[List["RiemannSiegelTheta", "[", "x", "]"]], "-", RowBox[List["x", " ", RowBox[List["Log", "[", "k", "]"]]]]]], "]"]], SqrtBox["k"]]]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["\[Nu]", "-", "1"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "\[Pi]"]], ")"]], RowBox[List["1", "/", "4"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["x", RowBox[List[RowBox[List["-", "1"]], "/", "4"]]], " ", RowBox[List["\[CapitalOmega]", "[", "p", "]"]]]], "-", FractionBox[RowBox[List[SuperscriptBox["x", RowBox[List[RowBox[List["-", "3"]], "/", "4"]]], " ", RowBox[List[SuperscriptBox["\[CapitalOmega]", TagBox[RowBox[List["(", "3", ")"]], Derivative], Rule[MultilineFunction, None]], "[", "p", "]"]]]], RowBox[List["48", " ", SqrtBox["2"], " ", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]]]]], "+", RowBox[List["2", " ", "\[Pi]", " ", SuperscriptBox["x", RowBox[List[RowBox[List["-", "5"]], "/", "4"]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[CapitalOmega]", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "p", "]"]], RowBox[List["64", " ", SuperscriptBox["\[Pi]", "2"]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[CapitalOmega]", TagBox[RowBox[List["(", "6", ")"]], Derivative], Rule[MultilineFunction, None]], "[", "p", "]"]], RowBox[List["18432", " ", SuperscriptBox["\[Pi]", "4"]]]]]], ")"]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "\[Pi]"]], ")"]], RowBox[List["3", "/", "2"]]], " ", SuperscriptBox["x", RowBox[List[RowBox[List["-", "7"]], "/", "4"]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[CapitalOmega]", "\[Prime]", Rule[MultilineFunction, None]], "[", "p", "]"]], RowBox[List["64", " ", SuperscriptBox["\[Pi]", "2"]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[CapitalOmega]", TagBox[RowBox[List["(", "5", ")"]], Derivative], Rule[MultilineFunction, None]], "[", "p", "]"]], RowBox[List["3840", " ", SuperscriptBox["\[Pi]", "4"]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[CapitalOmega]", TagBox[RowBox[List["(", "9", ")"]], Derivative], Rule[MultilineFunction, None]], "[", "p", "]"]], RowBox[List["5308416", " ", SuperscriptBox["\[Pi]", "6"]]]]]], ")"]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["\[Nu]", "\[Equal]", RowBox[List["Floor", "[", SqrtBox[FractionBox["x", RowBox[List["2", " ", "\[Pi]"]]]], "]"]]]], "&&", RowBox[List["p", "\[Equal]", RowBox[List[SqrtBox[FractionBox["x", RowBox[List["2", " ", "\[Pi]"]]]], "-", "\[Nu]"]]]], "&&", RowBox[List[RowBox[List["\[CapitalOmega]", "[", "p", "]"]], "\[Equal]", FractionBox[RowBox[List["Cos", "[", RowBox[List["2", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[SuperscriptBox["p", "2"], "-", "p", "-", FractionBox["1", "16"]]], ")"]]]], "]"]], RowBox[List["Cos", "[", RowBox[List["2", " ", "\[Pi]", " ", "p"]], "]"]]]]], "&&", RowBox[List["x", "\[Element]", "Reals"]], "&&", RowBox[List["(", RowBox[List["x", "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]]]










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





2001-10-29





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