(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 814217, 13409] NotebookOptionsPosition[ 812698, 13357] NotebookOutlinePosition[ 813367, 13381] CellTagsIndexPosition[ 813324, 13378] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Zeroes of the Zeta Function", "Subsubtitle"], Cell[TextData[{ "This handout illustrates ", StyleBox["Mathematica", FontSlant->"Italic"], "'s wide knowledge of mathematical functions.\n\nThe zeta function \ \[Zeta](n) for positive integer n is defined by" }], "Text"], Cell[BoxData[ RowBox[{ RowBox[{"\[Zeta]", RowBox[{"(", "n", ")"}]}], " ", "=", " ", RowBox[{ RowBox[{"1", " ", "+", " ", FractionBox["1", SuperscriptBox["2", "n"]], "+", FractionBox["1", SuperscriptBox["3", "n"]], "+", " ", FractionBox["1", SuperscriptBox["4", "n"]], "+"}], " ", "..."}]}]], "DisplayFormula"], Cell["\<\ It is well known that this blows up for n = 1 because the series is \ logarithmically divergent. Values of \[Zeta]{n) are known exactly for n an \ even positive integer, e.g.\ \>", "Text", CellChangeTimes->{3.419791550198553*^9}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Zeta", "[", "2", "]"}]], "Input", CellChangeTimes->{{3.514577686207024*^9, 3.514577686581834*^9}}], Cell[BoxData[ FractionBox[ SuperscriptBox["\[Pi]", "2"], "6"]], "Output", CellChangeTimes->{3.514577687325955*^9}] }, Open ]], Cell[TextData[{ "The definition given above also makes sense if one replaces n by complex z \ provided that Re[z] > 1 but blows up if Re[z] < 1. However, there are other \ definitions of \[Zeta](z), beyond the scope of the course, which agree with \ the above one where the latter is valid but which can be used ", StyleBox["anywhere", FontSlant->"Italic"], " in the complex plane (except for z=1 exactly where the function has a \ pole). \n\nThe properties of \[Zeta](z) in the complex plane, thus defined, \ have aroused a lot of interest, especially the location of the zeroes. There \ are \"trivial\" zeroes for z a negative even integer. This is illustrated in \ the plot below:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Plot", "[", RowBox[{ RowBox[{"Zeta", "[", "x", "]"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "10.5"}], ",", RowBox[{"-", "1"}]}], "}"}], ",", RowBox[{"PlotRange", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "10.5"}], ",", " ", "0"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "0.02"}], ",", "0.01"}], "}"}]}], "}"}]}], ",", RowBox[{"AxesLabel", "\[Rule]", RowBox[{"{", RowBox[{"\"\\"", ",", "\"\<\[Zeta](x)\>\""}], "}"}]}]}], "]"}]], "Input", CellChangeTimes->{{3.419791576740823*^9, 3.419791580145815*^9}, { 3.5145643090469017`*^9, 3.514564334091732*^9}}], Cell[BoxData[ GraphicsBox[{{}, {}, {Hue[0.67, 0.6, 0.6], PointSize[Large], Thickness[Large], LineBox[CompressedData[" 1:eJwlmnk4lO/3xy1jzzYzJstYZsZSqEhoUTfapGwRkn3fzRTZDWOpFEmKoiSl hKxFkvtokSKpTymlTaRkKUnK9n1+1++fmet1Off9nPN+n3Pm+QPDO2K3nwAf H98N4uP/vltuJMu7DKtDPPX69k9xaWgFid9pc4867D6xMjBlOw8VhKU2ZFxQ 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As an example, for the first root:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"y", " ", "/.", " ", RowBox[{"FindRoot", "[", RowBox[{ RowBox[{"Abs", "[", " ", RowBox[{"Zeta", "[", RowBox[{ RowBox[{"1", "/", "2"}], " ", "+", " ", RowBox[{"I", " ", "y"}]}], "]"}], "]"}], ",", " ", RowBox[{"{", RowBox[{"y", ",", " ", "14", ",", " ", "15"}], "}"}]}], "]"}]}]], "Input",\ CellChangeTimes->{{3.51432588515058*^9, 3.514325886537242*^9}}], Cell[BoxData["14.1347251417347`"], "Output", CellChangeTimes->{ 3.4197914473233137`*^9, 3.419792239601883*^9, 3.420194915902742*^9, 3.451691212792428*^9, 3.4517418336938353`*^9, 3.4829699477582617`*^9, { 3.514325874314426*^9, 3.514325887285536*^9}, 3.514564372988654*^9}] }, Open ]], Cell[TextData[{ "It can be shown that all zeroes but the trivial ones ", StyleBox["must", FontSlant->"Italic"], " lie the in the region 0 < Re[z] < 1. If you can find a zero in this region \ which does not have Re[z] = 1/2 you will be famous!\n\nBelow is a 3-D plot of \ 1/| \[Zeta](x +I y)| where the zeroes of \[Zeta] show up as peaks. The range \ covered is -2 < x < 2, 0 < y < 30. One sees the expected peaks at x = 1/2 \ (three of them for this range of y) and the peak corresponding to the trivial \ zero at x = -2, y = 0." }], "Text", CellChangeTimes->{{3.419791625792035*^9, 3.4197916351479397`*^9}, { 3.514578433892379*^9, 3.514578435047481*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Plot3D", "[", RowBox[{ FractionBox["1", RowBox[{"Abs", "[", RowBox[{"Zeta", "[", RowBox[{"x", "+", RowBox[{"\[ImaginaryI]", " ", "y"}]}], "]"}], "]"}]], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "1.999"}], ",", "2"}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", "0", ",", "30"}], "}"}], ",", RowBox[{"PlotPoints", "\[Rule]", "50"}], ",", RowBox[{"PlotRange", "\[Rule]", RowBox[{"{", RowBox[{"0", ",", "20"}], "}"}]}], ",", RowBox[{"ViewPoint", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"-", "30"}], ",", "10", ",", "7"}], "}"}]}], ",", RowBox[{"AxesLabel", "\[Rule]", RowBox[{"{", RowBox[{ "\"\\"", ",", "\"\\"", ",", "\"\<1/|\[Zeta](x+iy)|\>\""}], "}"}]}]}], "]"}]], "Input", 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