(* Content-type: application/vnd.wolfram.cdf.text *) (*** Wolfram CDF File ***) (* http://www.wolfram.com/cdf *) (* CreatedBy='Mathematica 9.0' *) (*************************************************************************) (* *) (* The Mathematica License under which this file was created prohibits *) (* restricting third parties in receipt of this file from republishing *) (* or redistributing it by any means, including but not limited to *) (* rights management or terms of use, without the express consent of *) (* Wolfram Research, Inc. 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FormBox["10", TraditionalForm]}, {20, FormBox["20", TraditionalForm]}, {30, FormBox["30", TraditionalForm]}}, {{0, FormBox["0", TraditionalForm]}, {1, FormBox["1", TraditionalForm]}, {2, FormBox["2", TraditionalForm]}, {3, FormBox["3", TraditionalForm]}, {4, FormBox["4", TraditionalForm]}}}, ViewPoint->{8, 1, 3}, ViewVertical->{0., 0., 1.}]], "Output"] }, {2}]], Cell[TextData[{ "En la siguiente representaci\[OAcute]n se muestra en un gr\[AAcute]fico \ polar de la funci\[OAcute]n zeta de Riemann a lo largo de la recta \ cr\[IAcute]tica ", Cell[BoxData[ RowBox[{" ", RowBox[{"s", "=", " ", RowBox[{ FractionBox["1", "2"], "+", RowBox[{"b", " ", "\[ImaginaryI]"}]}]}]}]]], ". Los ceros corresponden al pasa de la curva por el origen, puede verse \ que para el intervalo elegido (0, 26) ocurre cuatro veces." }], "Text"], Cell["\<\ Las coordendas polares corresponden a la parte real e imaginaria de \[Zeta](s)\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Manipulate", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"\[Zeta]", "[", "s_", "]"}], ":=", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"Re", "[", "#", "]"}], ",", RowBox[{"Im", "[", "#", "]"}]}], "}"}], "&"}], ")"}], "[", RowBox[{"Zeta", "[", RowBox[{"N", "[", "s", "]"}], "]"}], "]"}]}], ";", "\n", RowBox[{"ParametricPlot", "[", RowBox[{ RowBox[{"\[Zeta]", "[", RowBox[{ RowBox[{"1", "/", "2"}], "+", RowBox[{"t", " ", "I"}]}], "]"}], ",", " ", RowBox[{"{", RowBox[{"t", ",", " ", "0", ",", " ", "b"}], "}"}], ",", RowBox[{"AxesLabel", "\[Rule]", RowBox[{"{", RowBox[{ "\"\\"", ",", " ", "\"\\""}], "}"}]}]}], "]"}]}], ",", " ", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"b", ",", " ", "10"}], "}"}], ",", " ", "2", ",", " ", "40"}], "}"}], ",", RowBox[{"ContentSize", " ", "\[Rule]", " ", RowBox[{"{", RowBox[{"400", ",", "200"}], "}"}]}], ",", RowBox[{"SaveDefinitions", "\[Rule]", "True"}]}], "]"}]], "Input"], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`b$$ = 10, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{{ Hold[$CellContext`b$$], 10}, 2, 40}}, Typeset`size$$ = { 720., {155., 176.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = False, $CellContext`b$2770$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`b$$ = 10}, "ControllerVariables" :> { Hold[$CellContext`b$$, $CellContext`b$2770$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> ($CellContext`\[Zeta][ Pattern[$CellContext`s, Blank[]]] := ({ Re[#], Im[#]}& )[ Zeta[ N[$CellContext`s]]]; ParametricPlot[ $CellContext`\[Zeta][ 1/2 + $CellContext`t I], {$CellContext`t, 0, $CellContext`b$$}, AxesLabel -> { "Re(\!\(\*FractionBox[\(1\), \(2\)]\)+ b \[ImaginaryI])", "Im(\!\(\*FractionBox[\(1\), \(2\)]\)+ b \[ImaginaryI])"}]), "Specifications" :> {{{$CellContext`b$$, 10}, 2, 40}}, "Options" :> {ContentSize -> {400, 200}}, "DefaultOptions" :> {}], ImageSizeCache->{832., {260., 271.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, Initialization:>({$CellContext`\[Zeta][ Pattern[$CellContext`s, Blank[]]] := ({ Re[#], Im[#]}& )[ Zeta[ N[$CellContext`s]]], $CellContext`s = 0.5`13.698926085501514 + 14.13472514173469379091299960325094758351`15.150243448954702 I}; Typeset`initDone$$ = True), SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output"] }, {2}]], Cell[CellGroupData[{ Cell["\<\ La funci\[OAcute]n z y los n\[UAcute]meros primos\ \>", "Subsection"], Cell[TextData[{ "Una de las caracter\[IAcute]sticas principales de la funci\[OAcute]n z es \ su conexi\[OAcute]n con los n\[UAcute]meros primos.Recordemos que un n\ \[UAcute]mero primo ", StyleBox["p", FontSlant->"Italic"], " es aquel que solo es divisible por si mismo o por 1. El conjunto de los n\ \[UAcute]meros primos (que desde Euclides sabemos que es infinito) le \ llamamos \[WeierstrassP]. " }], "Text"], Cell[TextData[{ "Se puede demostrar que ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"\[Zeta]", "(", "s", ")"}], "\[LongEqual]", RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"k", "=", "1"}], "\[Infinity]"], FractionBox["1", SuperscriptBox["k", "s"]]}]}], TraditionalForm]], "InlineMath"], " , tambi\[EAcute]n puede obtenerse de la siguiente forma (", ButtonBox["http://es.wikipedia.org/wiki/Producto_de_Euler _para _la _funci\ \[OAcute]n _zeta _de _Riemann", BaseStyle->"Hyperlink", ButtonData->{ URL["http://es.wikipedia.org/wiki/Producto_de_Euler_para_la_funci\[OAcute]\ n_zeta_de_Riemann"], None}, ButtonNote-> "http://es.wikipedia.org/wiki/Producto_de_Euler_para_la_funci\[OAcute]n_\ zeta_de_Riemann"], "):" }], "Text"], Cell[TextData[Cell[BoxData[ FormBox[ TagBox[ RowBox[{ RowBox[{"\[Zeta]", "(", "s", ")"}], "=", RowBox[{ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"1", "-", FractionBox["1", SuperscriptBox["2", "s"]]}], ")"}], RowBox[{"(", RowBox[{"1", "-", FractionBox["1", SuperscriptBox["3", "s"]]}], ")"}], RowBox[{"(", RowBox[{"1", "-", FractionBox["1", SuperscriptBox["5", "s"]]}], ")"}]}], "+", "\[Ellipsis]"}], "=", RowBox[{ UnderscriptBox[ StyleBox["\[Product]", SpanMinSize->1., SpanMaxSize->1.], RowBox[{"p", "\[Element]", "\[WeierstrassP]"}]], FractionBox["1", RowBox[{"1", "-", SuperscriptBox["p", RowBox[{"-", "s"}]]}]]}]}]}], "MathMLPresentationTag", AutoDelete->True], TraditionalForm]]]], "Text"], Cell["\<\ Los n\[UAcute]meros primos aparentemente se distribuyen aleatoriamente. Si \ tenemos un n\[UAcute]mero primo no hay una formula (excepto verificar que el \ n\[UAcute]mero es primo) que permite determinar el siguiente. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Table", "[", RowBox[{ RowBox[{"Prime", "[", "n", "]"}], ",", RowBox[{"{", RowBox[{"n", ",", "45", ",", "60"}], "}"}]}], "]"}]], "Input", CellID->26752], Cell[BoxData[ RowBox[{"{", RowBox[{ "197", ",", "199", ",", "211", ",", "223", ",", "227", ",", "229", ",", "233", ",", "239", ",", "241", ",", "251", ",", "257", ",", "263", ",", "269", ",", "271", ",", "277", ",", "281"}], "}"}]], "Output", ImageSize->{479, 13}, ImageMargins->{{0, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}] }, Open ]], Cell["\<\ Cualquier n\[UAcute]mero natural se puede obtener de forma \[UAcute]nica como \ factores de numeros primos. En el ejemplo se eligen varios n\[UAcute]meros y \ se muestra su factores como n\[UAcute]meros primos, asi ocurre con cualquier \ n\[UAcute]mero.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"nums", "=", " ", RowBox[{"{", RowBox[{"1234", ",", "3281", ",", " ", "4352", ",", "5529"}], "}"}]}], ";"}], "\n", RowBox[{ RowBox[{"factores", "=", RowBox[{"Map", "[", RowBox[{"FactorInteger", ",", "nums"}], "]"}]}], ";"}], "\n", RowBox[{"Grid", "[", RowBox[{ RowBox[{"Transpose", "[", RowBox[{"{", RowBox[{"nums", ",", " ", RowBox[{ RowBox[{ RowBox[{"CenterDot", "@@", RowBox[{"(", RowBox[{"Superscript", "@@@", "#"}], ")"}]}], "&"}], "/@", " ", "factores"}]}], "}"}], "]"}], ",", RowBox[{"Frame", "\[Rule]", "All"}]}], "]"}]}], "Input"], Cell[BoxData[ TagBox[GridBox[{ {"1234", RowBox[{ TemplateBox[{"2","1"}, "Superscript"], "\[CenterDot]", TemplateBox[{"617","1"}, "Superscript"]}]}, {"3281", RowBox[{ TemplateBox[{"17","1"}, "Superscript"], "\[CenterDot]", 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FontSlant->"Italic"], " aumenta. " }], "Text"], Cell[TextData[{ "Hay varias aproximaciones para calcular \[Pi] (n):", StyleBox["t", FontSlant->"Italic"], "/Log(", StyleBox["t", FontSlant->"Italic"], "), o ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"li", "(", "n", ")"}], "=", RowBox[{ SubsuperscriptBox["\[Integral]", "0", "z"], RowBox[{ RowBox[{ RowBox[{"d", "t"}], "/", "log"}], " ", "t", " "}]}]}], TraditionalForm]], "InlineMath"], " (en ", StyleBox["Mathematica", FontSlant->"Italic"], " ", StyleBox["LogIntegral[x]", "Input"], "), pero la mejor es utilizar una funci\[OAcute]n basada en la z de \ Riemann. En concreto ", StyleBox["Mathematica", FontSlant->"Italic"], " utiliza la funci\[OAcute]n ", Cell[BoxData[ FormBox[ RowBox[{"R", RowBox[{"(", "n", ")"}]}], TraditionalForm]]], " =RiemannR[x]. " }], "Text"], Cell[TextData[{ "Para ", Cell[BoxData[ FormBox[ RowBox[{"x", ">", "0"}], TraditionalForm]], "InlineMath"], ", ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"R", "(", "x", ")"}], "\[LongEqual]", RowBox[{ UnderoverscriptBox["\[Sum]", "n", "\[Infinity]"], RowBox[{ RowBox[{ TagBox["\[Mu]", MoebiusMu], "(", "n", ")"}], " ", RowBox[{ RowBox[{"li", "(", SuperscriptBox["x", RowBox[{"1", "/", "n"}]], ")"}], "/", "n"}]}]}]}], TraditionalForm]], "InlineMath"], ". " }], "Text"], Cell["\<\ En la siguiente gr\[AAcute]fica comparamos el valor real de \[Pi](n) \ comparandolo con las estas aproximaciones. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Manipulate", "[", RowBox[{ RowBox[{"Plot", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"PrimePi", "[", "n", "]"}], ",", RowBox[{"n", "/", RowBox[{"Log", "[", "n", "]"}]}], ",", RowBox[{"LogIntegral", "[", "n", "]"}], ",", RowBox[{"RiemannR", "[", "n", "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{"n", ",", "1.5", ",", " ", "p"}], "}"}], ",", " ", RowBox[{"PlotLegends", "->", "\"\\""}]}], "]"}], ",", " ", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"p", ",", "10000"}], "}"}], ",", "100", ",", " ", RowBox[{"10", "^", "6"}]}], " ", "}"}]}], "]"}]], "Input", CellID->126584680], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`p$$ = 10000, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{{ Hold[$CellContext`p$$], 10000}, 100, 1000000}}, Typeset`size$$ = { 861., {208., 229.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, $CellContext`p$2827$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`p$$ = 10000}, "ControllerVariables" :> { Hold[$CellContext`p$$, $CellContext`p$2827$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> Plot[{ PrimePi[$CellContext`n], $CellContext`n/Log[$CellContext`n], LogIntegral[$CellContext`n], RiemannR[$CellContext`n]}, {$CellContext`n, 1.5, $CellContext`p$$}, PlotLegends -> "Expressions"], "Specifications" :> {{{$CellContext`p$$, 10000}, 100, 1000000}}, "Options" :> {}, "DefaultOptions" :> {}], ImageSizeCache->{951., {301., 312.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", ImageSize->{184, 118}, ImageMargins->{{0, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}] }, {2}]], Cell[TextData[{ "Puede visualizarse mejor si representamos la diferencia entre las distintas \ funciones y \[Pi](", StyleBox["n", FontSlant->"Italic"], "). 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