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 RiemannSiegelZ

 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

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["RiemannSiegelZ", "[", "x", "]"]], "\[Proportional]", 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", "]"]]]], "-", RowBox[List[FractionBox[SuperscriptBox["x", RowBox[List[RowBox[List["-", "3"]], "/", "4"]]], RowBox[List["48", " ", SqrtBox["2"], " ", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]]]]], RowBox[List[SuperscriptBox["\[CapitalOmega]", TagBox[RowBox[List["(", "3", ")"]], Derivative], Rule[MultilineFunction, None]], "[", "p", "]"]]]], "+", 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]"]]]], "]"]]]], "\[And]", RowBox[List["p", "\[Equal]", RowBox[List[SqrtBox[FractionBox["x", RowBox[List["2", " ", "\[Pi]"]]]], "-", "\[Nu]"]]]], "\[And]", RowBox[List[RowBox[List["\[CapitalOmega]", "[", "p", "]"]], "\[Equal]", FractionBox[RowBox[List["Cos", "[", RowBox[List["2", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[SuperscriptBox["p", "2"], "-", "p", "-", RowBox[List["1", "/", "16"]]]], ")"]]]], "]"]], RowBox[List["Cos", "[", RowBox[List["2", " ", "\[Pi]", " ", "p"]], "]"]]]]], " ", "\[And]", RowBox[List["x", "\[Element]", "Reals"]], "\[And]", RowBox[List["(", RowBox[List["x", "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]

 MathML Form

 Z TagBox["Z", RiemannSiegelZ] ( x ) 2 k = 1 ν cos ( ϑ TagBox["\[CurlyTheta]", RiemannSiegelTheta] ( x ) - x log ( k ) ) k + ( - 1 ) ν - 1 2 π 4 ( Ω ( p ) x 4 - Ω ( 3 ) TagBox[RowBox[List["(", "3", ")"]], Derivative] ( p ) 48 2 π 3 / 2 x - 3 / 4 + 2 π ( Ω ′′ ( p ) 64 π 2 + Ω ( 6 ) TagBox[RowBox[List["(", "6", ")"]], Derivative] ( p ) 18432 π 4 ) x - 5 / 4 - ( 2 π ) 3 / 2 ( Ω ( p ) 64 π 2 + Ω ( 5 ) TagBox[RowBox[List["(", "5", ")"]], Derivative] ( p ) 3840 π 4 + Ω ( 9 ) TagBox[RowBox[List["(", "9", ")"]], Derivative] ( p ) 5308416 π 6 ) x - 7 / 4 ) /; ν x 2 π p x 2 π - ν Ω ( p ) 1 cos ( 2 π p ) cos ( 2 π ( p 2 - p - 1 16 ) ) x TagBox["\[DoubleStruckCapitalR]", Function[Reals]] ( x "\[Rule]" ) Condition Proportional RiemannSiegelZ x 2 k 1 ν RiemannSiegelTheta x -1 x k k 1 2 -1 -1 ν -1 2 1 4 Ω p x 1 4 -1 -1 p 3 Ω p 48 2 1 2 3 2 -1 x -3 4 2 p 2 Ω p 64 2 -1 p 6 Ω p 18432 4 -1 x -5 4 -1 2 3 2 p Ω p 64 2 -1 p 5 Ω p 3840 4 -1 p 9 Ω p 5308416 6 -1 x -7 4 ν x 2 -1 1 2 p x 2 -1 1 2 -1 ν Ω p 1 2 p -1 2 p 2 -1 p -1 1 16 x Rule x [/itex]

 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