(* 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[ 132398, 3591] NotebookOptionsPosition[ 91393, 2614] NotebookOutlinePosition[ 123209, 3337] CellTagsIndexPosition[ 123129, 3332] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["\<\ Computing Knopfmacher's limit, or: My first foray into computational \ mathematics, reprise\ \>", "Title", CellChangeTimes->{{3.407166903767702*^9, 3.407166948062443*^9}, 3.407175662380329*^9, 3.42512660217809*^9, {3.454760004538085*^9, 3.454760014814987*^9}}], Cell["\<\ \[ThinSpace]Daniel Lichtblau danl@wolfram.com Wolfram Research, Inc. 100 Trade Center Dr. Champaign IL 61820\ \>", "Author", CellChangeTimes->{{3.424788950876697*^9, 3.42478895850612*^9}, 3.4260820382074814`*^9}], Cell["\<\ ACA 2009 Montreal, Canada Session: Applications of Math Software to Mathematical Research \ \>", "Subtitle", CellChangeTimes->{{3.424788899331741*^9, 3.424788899843505*^9}, { 3.424788931351298*^9, 3.424788964289202*^9}, {3.424789021912471*^9, 3.424789035006227*^9}, 3.425126650473429*^9, 3.425126686186935*^9, 3.426082042794077*^9, {3.454759943950425*^9, 3.454759954157041*^9}, 3.4634232740416603`*^9, 3.4634233142946663`*^9}] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell[TextData[StyleBox["ABSTRACT", FontWeight->"Bold"]], "Subsection", CellChangeTimes->{{3.424789104977*^9, 3.424789114292745*^9}}], Cell["\<\ I will discuss a problem I encountered over a decade ago, and worked on via \ internet with someone I (alas) never met. It involves a mix of number theory, \ real analysis, hard-core computation, and some slightly perplexing results. In brief, we begin with a function expressed as a certain infinite product; \ Arnold Knopfmacher encountered it in an attempt to approximate the number of \ irreducible factors of univariate polynomials over Galois fields and raised \ the question of how to obtain a certain limit to this function. We derive and \ execute an effective algorithm for the task at hand. We'll also indicate why \ the most \"obvious\" approach does not work well in practice, or at all in \ theory.\ \>", "Abstract", CellChangeTimes->{ 3.407167031516175*^9, {3.407167383675258*^9, 3.407167570417365*^9}, { 3.40717004029304*^9, 3.407170093516818*^9}, 3.424789108648882*^9, { 3.425126699734592*^9, 3.425126748559512*^9}, {3.454760026919868*^9, 3.4547601052315598`*^9}}], Cell[TextData[{ ButtonBox["\[FilledLeftTriangle]\[ThickSpace]\[ThickSpace]\[ThickSpace]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPagePrevious"]}], ButtonNote->FEPrivate`FrontEndResource[ "FEStrings", "SlideshowPrevSlideText"]], "\[ThickSpace]\[ThickSpace]|\[ThickSpace]\[ThickSpace]", ButtonBox["\[ThickSpace]\[ThickSpace]\[ThickSpace]\[FilledRightTriangle]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageNext"]}], ButtonNote->FEPrivate`FrontEndResource[ "FEStrings", "SlideshowNextSlideText"]] }], "PreviousNext"] }, Open ]] }, Open ]], Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["The problem", "Section", CellChangeTimes->{{3.424789104977*^9, 3.424789114292745*^9}, 3.424789299633628*^9, {3.42479274300571*^9, 3.424792772016237*^9}, { 3.424794212776393*^9, 3.424794244645659*^9}, {3.425078783064349*^9, 3.425078784077486*^9}, {3.425131398588085*^9, 3.425131402972484*^9}, 3.4260907728072195`*^9, {3.454761944638286*^9, 3.454761946812891*^9}, { 3.454762798051714*^9, 3.454762815939533*^9}}], Cell["\<\ As posed by Arnold Knopfmacher to the Usenet group \ comp.soft-sys.math.mathematica in January 1999\ \>", "Subsubsection", CellChangeTimes->{{3.424789104977*^9, 3.424789114292745*^9}, 3.424789299633628*^9, {3.42479274300571*^9, 3.424792772016237*^9}, { 3.424794212776393*^9, 3.424794244645659*^9}, {3.425078783064349*^9, 3.425078784077486*^9}, {3.425131398588085*^9, 3.425131402972484*^9}, 3.4260907728072195`*^9, {3.454761944638286*^9, 3.454761946812891*^9}, { 3.454762798051714*^9, 3.4547628689004087`*^9}}], Cell[TextData[{ "Let ", Cell[BoxData[ FormBox[ RowBox[{"d", "(", "k", ")"}], TraditionalForm]], FormatType->"TraditionalForm"], " denote the smallest prime factor of ", Cell[BoxData[ FormBox["k", TraditionalForm]], FormatType->"TraditionalForm"], "." }], "Text", CellChangeTimes->{{3.454762106798238*^9, 3.454762132511546*^9}}], Cell[TextData[{ "Define ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"m", RowBox[{"(", "k", ")"}]}], "=", RowBox[{"k", " ", "-", " ", RowBox[{"k", "/", RowBox[{"d", "(", "k", ")"}]}]}]}], TraditionalForm]], FormatType->"TraditionalForm"] }], "Text", CellChangeTimes->{{3.454761960301345*^9, 3.454761963196878*^9}, { 3.454762051934536*^9, 3.4547620528617153`*^9}, {3.4547620857580557`*^9, 3.4547620999698963`*^9}, {3.45476214149434*^9, 3.454762143254472*^9}}], Cell["Define", "Text", CellChangeTimes->{{3.454762151774371*^9, 3.454762152526638*^9}}], Cell[BoxData[ StyleBox[Cell[TextData[Cell[BoxData[ FormBox[ StyleBox[ RowBox[{ RowBox[{"p", "(", "x", ")"}], "=", FractionBox[ RowBox[{ UnderoverscriptBox[ StyleBox["\[Product]", FontFamily->"Times"], RowBox[{"k", "=", "2"}], "\[Infinity]"], StyleBox[ RowBox[{"(", RowBox[{"1", "-", FractionBox[ SuperscriptBox["x", RowBox[{"m", "(", "k", ")"}]], RowBox[{"k", "+", "1"}]]}], ")"}], FontFamily->"Times"]}], RowBox[{"1", "-", "x"}]]}], FontFamily->"Times"], TraditionalForm]]]]], FontFamily->"Times", FontWeight->"Plain"]], "DisplayFormula", CellChangeTimes->{{3.454761967713358*^9, 3.4547620047345057`*^9}, { 3.454762047387002*^9, 3.4547620834506483`*^9}, {3.454762157252261*^9, 3.454762213868279*^9}, {3.45476248779401*^9, 3.454762495485602*^9}}], Cell["\<\ We wish to compute numerically, to at least eight decimal places, the \ following limit.\ \>", "Text", CellChangeTimes->{{3.4547622896158743`*^9, 3.4547622949435577`*^9}, { 3.454762748403493*^9, 3.454762766995008*^9}}], Cell[BoxData[ StyleBox[Cell[TextData[{ Cell[BoxData[ FormBox[ UnderscriptBox["lim", RowBox[{"x", "\[RightArrow]", SuperscriptBox["1", "-"]}]], TraditionalForm]], FormatType->"TraditionalForm"], " ", Cell[BoxData[ FormBox[ StyleBox[ RowBox[{"p", "(", "x", ")"}], FontFamily->"Times"], TraditionalForm]]] }]], FontFamily->"Times", FontWeight->"Plain"]], "DisplayFormula", CellChangeTimes->{ 3.4547622993657618`*^9, {3.454762336197165*^9, 3.4547624272387457`*^9}, { 3.454762512046627*^9, 3.4547625373180113`*^9}, {3.454762579445212*^9, 3.45476257984482*^9}, 3.4547659357911673`*^9}], Cell[TextData[{ "That is to say, we compute the limit as ", Cell[BoxData[ FormBox["x", TraditionalForm]], FormatType->"TraditionalForm"], " approaches 1 from the left (lesser) side." }], "Text", CellChangeTimes->{{3.454762548881912*^9, 3.454762569361863*^9}, { 3.4547659402700043`*^9, 3.45476594454062*^9}}], Cell[TextData[{ ButtonBox["\[FilledLeftTriangle]\[ThickSpace]\[ThickSpace]\[ThickSpace]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPagePrevious"]}], ButtonNote->FEPrivate`FrontEndResource[ "FEStrings", "SlideshowPrevSlideText"]], "\[ThickSpace]\[ThickSpace]|\[ThickSpace]\[ThickSpace]", ButtonBox["\[ThickSpace]\[ThickSpace]\[ThickSpace]\[FilledRightTriangle]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageNext"]}], ButtonNote->FEPrivate`FrontEndResource[ "FEStrings", "SlideshowNextSlideText"]] }], "PreviousNext"] }, Open ]], Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Why do we care?", "Section", CellChangeTimes->{{3.424789104977*^9, 3.424789114292745*^9}, 3.424789299633628*^9, {3.42479274300571*^9, 3.424792772016237*^9}, { 3.424792855532515*^9, 3.424792857438486*^9}, {3.424792942847715*^9, 3.424792946783374*^9}, {3.425078775130708*^9, 3.425078776673465*^9}, 3.426090782441072*^9, {3.454762724756665*^9, 3.454762728339218*^9}}], Cell["\<\ There are similar formulas in a paper from 1995 by Knopfmacher and Warlimont, \ analyzing probabilities related to numbers of irreducible factors of \ distinct degrees in univariate polynomials over Galois fields\ \>", "Item", CellChangeTimes->{{3.424792952139543*^9, 3.424792957022748*^9}, { 3.424793062139309*^9, 3.424793071907915*^9}, {3.424793199640255*^9, 3.424793254242299*^9}, {3.424793506836782*^9, 3.424793513969859*^9}, { 3.42513168100663*^9, 3.425131681014843*^9}, 3.4547627327075787`*^9, 3.4547627900421057`*^9, {3.454762879204665*^9, 3.454762893604618*^9}}], Cell["It's an interesting computation", "Item", CellChangeTimes->{{3.425131681942174*^9, 3.425131702980809*^9}, { 3.454762915109013*^9, 3.4547629201966047`*^9}}], Cell[TextData[{ ButtonBox["\[FilledLeftTriangle]\[ThickSpace]\[ThickSpace]\[ThickSpace]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPagePrevious"]}], ButtonNote->FEPrivate`FrontEndResource[ "FEStrings", "SlideshowPrevSlideText"]], "\[ThickSpace]\[ThickSpace]|\[ThickSpace]\[ThickSpace]", ButtonBox["\[ThickSpace]\[ThickSpace]\[ThickSpace]\[FilledRightTriangle]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageNext"]}], ButtonNote->FEPrivate`FrontEndResource[ "FEStrings", "SlideshowNextSlideText"]] }], "PreviousNext"] }, Open ]], Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["My history with this problem", "Section", CellChangeTimes->{{3.424789104977*^9, 3.424789114292745*^9}, 3.424789299633628*^9, {3.42479274300571*^9, 3.424792772016237*^9}, { 3.424792855532515*^9, 3.424792857438486*^9}, {3.424792942847715*^9, 3.424792946783374*^9}, {3.425078775130708*^9, 3.425078776673465*^9}, { 3.425132282397209*^9, 3.425132316907309*^9}, {3.425133267637193*^9, 3.425133291279485*^9}, 3.426090800036373*^9, {3.454766080927012*^9, 3.454766085741762*^9}}], Cell["\<\ I worked on it off and on for several days. Then someone else reading the \ forum contributed a similar result, but much more precise. His name was J\ \[UDoubleDot]rgen Tischer, a math department faculty member of Universidad \ del Valle, Columbia. We corresponded a bit over a period of months, and I \ wrote up the results. I lost contact with him a year or two later. I had \ wanted to invite him to this conference. After getting nowhere with an \ internet search for a current contact address, I learned he had passed away \ in January of 2008. I felt it fitting to talk a bit about this problem, since \ it used ideas of his and also was one of my first forays into computational \ mathematics.\ \>", "Text", CellChangeTimes->{{3.425133920971047*^9, 3.425133994056337*^9}, { 3.425134085748006*^9, 3.425134095495784*^9}, {3.425134153821338*^9, 3.425134281670207*^9}, {3.425134315924928*^9, 3.425134337565814*^9}, 3.454766117774639*^9, {3.4547661976790953`*^9, 3.454766424464862*^9}, { 3.454767773884513*^9, 3.454767785980204*^9}}] }, Open ]], Cell[TextData[{ ButtonBox["\[FilledLeftTriangle]\[ThickSpace]\[ThickSpace]\[ThickSpace]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPagePrevious"]}], ButtonNote->FEPrivate`FrontEndResource[ "FEStrings", "SlideshowPrevSlideText"]], "\[ThickSpace]\[ThickSpace]|\[ThickSpace]\[ThickSpace]", ButtonBox["\[ThickSpace]\[ThickSpace]\[ThickSpace]\[FilledRightTriangle]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageNext"]}], ButtonNote->FEPrivate`FrontEndResource[ "FEStrings", "SlideshowNextSlideText"]] }], "PreviousNext"], Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Candidate results", "Section", CellChangeTimes->{{3.424789104977*^9, 3.424789114292745*^9}, 3.424789299633628*^9, {3.42479274300571*^9, 3.424792772016237*^9}, { 3.424792855532515*^9, 3.424792857438486*^9}, {3.424792942847715*^9, 3.424792946783374*^9}, {3.425078775130708*^9, 3.425078776673465*^9}, { 3.425132282397209*^9, 3.425132316907309*^9}, 3.4260907856657085`*^9, { 3.454765012006781*^9, 3.4547650173654213`*^9}, {3.454786208897409*^9, 3.454786213823757*^9}}], Cell["\<\ There were four responses, with two (mine and Tischer's) giving roughly the \ same results. The proposed values were\ \>", "Text", CellChangeTimes->{{3.454765245319281*^9, 3.4547652792395163`*^9}, { 3.454766469248938*^9, 3.454766492849012*^9}}], Cell["1.3397", "Item", CellChangeTimes->{{3.454765145213756*^9, 3.454765148407894*^9}, { 3.454765228331564*^9, 3.45476522997514*^9}, 3.454765306791338*^9}], Cell["2 (exactly)", "Item", CellChangeTimes->{{3.454765145213756*^9, 3.454765148407894*^9}, { 3.454765228331564*^9, 3.45476522997514*^9}, {3.454765306791338*^9, 3.454765313976054*^9}, {3.4547862171681767`*^9, 3.454786220927846*^9}}], Cell["2.292", "Item", CellChangeTimes->{{3.454765145213756*^9, 3.454765148407894*^9}, { 3.454765228331564*^9, 3.45476522997514*^9}, {3.454765306791338*^9, 3.4547653127921267`*^9}, 3.4547654027450314`*^9}], Cell["So which is correct?", "Text", CellChangeTimes->{{3.454765406728458*^9, 3.454765444088537*^9}}], Cell["More important: How do we even know the limit exists?", "Text", CellChangeTimes->{{3.454765406728458*^9, 3.454765448568884*^9}}], Cell[TextData[{ ButtonBox["\[FilledLeftTriangle]\[ThickSpace]\[ThickSpace]\[ThickSpace]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPagePrevious"]}], ButtonNote->FEPrivate`FrontEndResource[ "FEStrings", "SlideshowPrevSlideText"]], "\[ThickSpace]\[ThickSpace]|\[ThickSpace]\[ThickSpace]", ButtonBox["\[ThickSpace]\[ThickSpace]\[ThickSpace]\[FilledRightTriangle]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageNext"]}], ButtonNote->FEPrivate`FrontEndResource[ "FEStrings", "SlideshowNextSlideText"]] }], "PreviousNext"] }, Open ]], Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Easy to show...", "Section", CellChangeTimes->{{3.424789104977*^9, 3.424789114292745*^9}, 3.424789299633628*^9, {3.42479274300571*^9, 3.424792772016237*^9}, { 3.424792855532515*^9, 3.424792857438486*^9}, {3.424792942847715*^9, 3.424792946783374*^9}, {3.425078775130708*^9, 3.425078776673465*^9}, { 3.425132282397209*^9, 3.425132316907309*^9}, {3.425132508657257*^9, 3.4251325266765*^9}, 3.426090790362462*^9, {3.454765485402937*^9, 3.4547654896571093`*^9}}], Cell[TextData[{ "A ", StyleBox["lim sup", FontSlant->"Italic"], " and ", StyleBox["lim inf ", FontSlant->"Italic"], " both are readily computed." }], "Item", CellChangeTimes->{{3.425132559502735*^9, 3.42513261074816*^9}, { 3.425132796702665*^9, 3.425132817315332*^9}, {3.425170520078504*^9, 3.425170528980835*^9}, {3.425171160381882*^9, 3.425171161585722*^9}, { 3.454765495306782*^9, 3.454765503705368*^9}, {3.4547655353217287`*^9, 3.4547655429057198`*^9}, {3.4547656061860657`*^9, 3.4547656061889057`*^9}, { 3.4547862361455593`*^9, 3.4547862525123672`*^9}}] }, Open ]], Cell[TextData[{ "A ", StyleBox["lim inf ", FontSlant->"Italic"], " is given by ", Cell[BoxData[ FormBox[ SuperscriptBox["\[ExponentialE]", "\[Gamma]"], TraditionalForm]], FormatType->"TraditionalForm"], "(the exponential of the Euler gamma constant). This is around 1.871, \ so...1.3397 will exit stage left." }], "Item", CellChangeTimes->{{3.454765607146187*^9, 3.4547656766937723`*^9}, { 3.454765711582473*^9, 3.454765746555664*^9}, {3.454765791904232*^9, 3.4547657947393923`*^9}, {3.4547658919488277`*^9, 3.4547658996926413`*^9}, { 3.454765984799903*^9, 3.454765985663028*^9}}], Cell[TextData[{ ButtonBox["\[FilledLeftTriangle]\[ThickSpace]\[ThickSpace]\[ThickSpace]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPagePrevious"]}], ButtonNote->FEPrivate`FrontEndResource[ "FEStrings", "SlideshowPrevSlideText"]], "\[ThickSpace]\[ThickSpace]|\[ThickSpace]\[ThickSpace]", ButtonBox["\[ThickSpace]\[ThickSpace]\[ThickSpace]\[FilledRightTriangle]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageNext"]}], ButtonNote->FEPrivate`FrontEndResource[ "FEStrings", "SlideshowNextSlideText"]] }], "PreviousNext"], Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell[TextData[{ "Definitions in ", StyleBox["Mathematica", FontSlant->"Italic"] }], "Section", CellChangeTimes->{{3.424789104977*^9, 3.424789114292745*^9}, 3.424789299633628*^9, {3.42479274300571*^9, 3.424792772016237*^9}, { 3.424792855532515*^9, 3.424792857438486*^9}, {3.424792942847715*^9, 3.424792946783374*^9}, {3.425078775130708*^9, 3.425078776673465*^9}, { 3.425132282397209*^9, 3.425132316907309*^9}, {3.425132922745557*^9, 3.425132936346845*^9}, 3.4260907967416353`*^9, {3.454766047967881*^9, 3.454766054413602*^9}, {3.4547665642950363`*^9, 3.454766565393722*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{"d", "[", "k_", "]"}], " ", ":=", " ", RowBox[{ RowBox[{"Divisors", "[", "k", "]"}], "[", RowBox[{"[", "2", "]"}], "]"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{"m", "[", "k_", "]"}], ":=", RowBox[{"k", "-", FractionBox["k", RowBox[{"d", "[", "k", "]"}]]}]}], "\n", RowBox[{ RowBox[{"p", "[", "x_", "]"}], ":=", FractionBox[ RowBox[{ UnderoverscriptBox["\[Product]", RowBox[{"k", "=", "2"}], "\[Infinity]"], RowBox[{"(", RowBox[{"1", "-", FractionBox[ SuperscriptBox["x", RowBox[{"m", "[", "k", "]"}]], RowBox[{"k", "+", "1"}]]}], ")"}]}], RowBox[{"1", "-", "x"}]]}]}], "Input"], Cell["General idea: Start with", "Text", CellChangeTimes->{{3.454766611409813*^9, 3.454766613761817*^9}, { 3.454766912245453*^9, 3.4547669146923428`*^9}}], Cell[TextData[{ " ", Cell[BoxData[ StyleBox[Cell[TextData[Cell[BoxData[ FormBox[ StyleBox[ RowBox[{ FractionBox[ RowBox[{ UnderoverscriptBox["\[Product]", RowBox[{"k", "=", "2"}], "\[Infinity]"], RowBox[{"(", RowBox[{"1", "-", FractionBox[ SuperscriptBox["x", RowBox[{"\[LeftCeiling]", FractionBox["k", "2"], "\[RightCeiling]"}]], RowBox[{"k", "+", "1"}]]}], ")"}]}], RowBox[{"1", "-", "x"}]], "<", RowBox[{"p", "[", "x", "]"}], "<", FractionBox[ RowBox[{ UnderoverscriptBox["\[Product]", RowBox[{"k", "=", "2"}], "\[Infinity]"], RowBox[{"(", RowBox[{"1", "-", FractionBox[ SuperscriptBox["x", RowBox[{"k", "-", "1"}]], RowBox[{"k", "+", "1"}]]}], ")"}]}], RowBox[{"1", "-", "x"}]]}], FontFamily->"Times", FontWeight->"Plain"], TraditionalForm]]]]], FontFamily->"Times", FontWeight->"Plain"]], "DisplayFormula"] }], "InlineFormula", CellChangeTimes->{{3.425132677209151*^9, 3.425132777269512*^9}, { 3.425133089145393*^9, 3.42513309738645*^9}, 3.454766172591078*^9, { 3.454766582117275*^9, 3.454766606645464*^9}}], Cell[TextData[{ "Now take logs to get summations. Expand logs at 1 as power series, \ obtaining double summations. Switch order of summation (requires \ justification), and we find that log of a ", StyleBox["lim sup", FontSlant->"Italic"], " is ", Cell[BoxData[ FormBox[ RowBox[{"\[Gamma]", " ", "+", " ", RowBox[{"log", RowBox[{"(", "2", ")"}]}]}], TraditionalForm]], FormatType->"TraditionalForm"], ". Finding a ", StyleBox["lim inf", FontSlant->"Italic"], " is similar though a bit more work." }], "Text", CellChangeTimes->{{3.4547666370899754`*^9, 3.454766641506049*^9}, { 3.454766675730425*^9, 3.454766707618697*^9}, {3.4547668139557133`*^9, 3.4547669526776333`*^9}, {3.454767043349661*^9, 3.454767067398953*^9}}], Cell[TextData[{ ButtonBox["\[FilledLeftTriangle]\[ThickSpace]\[ThickSpace]\[ThickSpace]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPagePrevious"]}], ButtonNote->FEPrivate`FrontEndResource[ "FEStrings", "SlideshowPrevSlideText"]], "\[ThickSpace]\[ThickSpace]|\[ThickSpace]\[ThickSpace]", ButtonBox["\[ThickSpace]\[ThickSpace]\[ThickSpace]\[FilledRightTriangle]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageNext"]}], ButtonNote->FEPrivate`FrontEndResource[ "FEStrings", "SlideshowNextSlideText"]] }], "PreviousNext"] }, Open ]], Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell["\<\ What has changed in the past decade?\ \>", "Section", CellChangeTimes->{{3.424789104977*^9, 3.424789114292745*^9}, 3.424789299633628*^9, {3.42479274300571*^9, 3.424792772016237*^9}, { 3.424792855532515*^9, 3.424792857438486*^9}, {3.424792942847715*^9, 3.424792946783374*^9}, {3.425078775130708*^9, 3.425078776673465*^9}, { 3.425132282397209*^9, 3.425132316907309*^9}, {3.425133267637193*^9, 3.425133291279485*^9}, 3.426090800036373*^9, {3.454766080927012*^9, 3.454766085741762*^9}, {3.454767078679344*^9, 3.454767086789691*^9}}], Cell["\<\ Ten years ago this computing took manual intervention. I had to do things \ like split sums, and do further contortions to take limits. Today some can be \ done directly. Here is one such that arose in the process.\ \>", "Text", CellChangeTimes->{{3.454767095029675*^9, 3.454767117750677*^9}, { 3.454767212439254*^9, 3.4547672145348043`*^9}, {3.454767335400136*^9, 3.454767381592058*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Limit", "[", RowBox[{ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"k", "=", "3"}], "M"], RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"j", "=", "2"}], "\[Infinity]"], FractionBox["1", RowBox[{"j", " ", SuperscriptBox["k", "j"]}]]}]}], ",", RowBox[{"M", "\[Rule]", "\[Infinity]"}]}], "]"}]], "Input", CellChangeTimes->{{3.454767022234207*^9, 3.4547670315468493`*^9}}], Cell[BoxData[ RowBox[{ FractionBox["3", "2"], "-", "EulerGamma", "-", RowBox[{"Log", "[", "2", "]"}]}]], "Output", CellChangeTimes->{3.4547677504420967`*^9}] }, Open ]], Cell[TextData[{ ButtonBox["\[FilledLeftTriangle]\[ThickSpace]\[ThickSpace]\[ThickSpace]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPagePrevious"]}], ButtonNote->FEPrivate`FrontEndResource[ "FEStrings", "SlideshowPrevSlideText"]], "\[ThickSpace]\[ThickSpace]|\[ThickSpace]\[ThickSpace]", ButtonBox["\[ThickSpace]\[ThickSpace]\[ThickSpace]\[FilledRightTriangle]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageNext"]}], ButtonNote->FEPrivate`FrontEndResource[ "FEStrings", "SlideshowNextSlideText"]] }], "PreviousNext"], Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["A start at approximating the actual limit", "Section", CellChangeTimes->{{3.424789104977*^9, 3.424789114292745*^9}, 3.424789299633628*^9, {3.42479274300571*^9, 3.424792772016237*^9}, { 3.424792855532515*^9, 3.424792857438486*^9}, {3.424792942847715*^9, 3.424792946783374*^9}, {3.425078775130708*^9, 3.425078776673465*^9}, { 3.425132282397209*^9, 3.425132316907309*^9}, {3.425134477560227*^9, 3.425134491746237*^9}, 3.426090805954883*^9, {3.454767820876114*^9, 3.4547678304916687`*^9}}], Cell["Truncate the series for the logarithm", "Item", CellChangeTimes->{{3.4547678325796337`*^9, 3.454767883019609*^9}}], Cell[TextData[{ "Evaluate using exact or high precision arithmetic at ", Cell[BoxData[ FormBox[ RowBox[{"x", "=", "1"}], TraditionalForm]], "DisplayFormulaInline"] }], "Item", CellChangeTimes->{{3.4547678325796337`*^9, 3.45476789098252*^9}}], Cell["Exponentiate", "Item", CellChangeTimes->{{3.4547678325796337`*^9, 3.454767898380323*^9}}], Cell[BoxData[ StyleBox[ RowBox[{ RowBox[{"Log", "[", RowBox[{"p", "[", "x", "]"}], "]"}], "=", RowBox[{ RowBox[{ RowBox[{ RowBox[{"-", RowBox[{"Log", "[", RowBox[{"1", "-", "x"}], "]"}]}], "+", RowBox[{ UnderscriptBox["\[Sum]", RowBox[{"2", "|", "k"}]], RowBox[{"Log", "[", RowBox[{"1", "-", FractionBox[ SuperscriptBox["x", RowBox[{"k", "/", "2"}]], RowBox[{"k", "+", "1"}]]}], "]"}]}], "+", RowBox[{ UnderscriptBox["\[Sum]", RowBox[{ RowBox[{"2", "\[NotVerticalBar]", "k"}], ",", RowBox[{"3", "|", "k"}]}]], RowBox[{"Log", "[", RowBox[{"1", "-", FractionBox[ SuperscriptBox["x", RowBox[{"2", " ", RowBox[{"k", "/", "3"}]}]], RowBox[{"k", "+", "1"}]]}], "]"}]}], "+"}], "..."}], "=", RowBox[{ RowBox[{ RowBox[{"-", RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"k", "=", "1"}], "\[Infinity]"], FractionBox[ SuperscriptBox["x", "k"], "k"]}]}], "+", RowBox[{ UnderscriptBox["\[Sum]", RowBox[{"2", "|", "k"}]], FractionBox[ SuperscriptBox["x", RowBox[{"k", "/", "2"}]], RowBox[{"k", "+", "1"}]]}], "+", RowBox[{ UnderscriptBox["\[Sum]", RowBox[{"2", "|", "k"}]], RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"j", "=", "2"}], "\[Infinity]"], FractionBox[ SuperscriptBox["x", RowBox[{"j", " ", RowBox[{"k", "/", "2"}]}]], RowBox[{"j", " ", SuperscriptBox[ RowBox[{"(", RowBox[{"k", "+", "1"}], ")"}], "j"]}]]}]}], "+", RowBox[{ UnderscriptBox["\[Sum]", RowBox[{ RowBox[{"2", "\[NotVerticalBar]", "k"}], ",", RowBox[{"3", "|", "k"}]}]], FractionBox[ SuperscriptBox["x", RowBox[{"2", " ", RowBox[{"k", "/", "3"}]}]], RowBox[{"k", "+", "1"}]]}], "+", RowBox[{ UnderscriptBox["\[Sum]", RowBox[{ RowBox[{"2", "\[NotVerticalBar]", "k"}], ",", RowBox[{"3", "|", "k"}]}]], RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"j", "=", "2"}], "\[Infinity]"], FractionBox[ SuperscriptBox["x", RowBox[{ RowBox[{"(", RowBox[{"2", "/", "3"}], ")"}], "j", " ", "k"}]], RowBox[{"j", " ", SuperscriptBox[ RowBox[{"(", RowBox[{"k", "+", "1"}], ")"}], "j"]}]]}]}], " ", "+"}], "..."}]}]}], FontFamily->"Times", FontWeight->"Plain"]], "DisplayFormula", CellChangeTimes->{{3.454768115671893*^9, 3.454768146673031*^9}}], Cell[TextData[{ "(The series that get truncated are the summations in", Cell[BoxData[ FormBox[ RowBox[{" ", "j"}], TraditionalForm]], FormatType->"TraditionalForm"], "). From this tactic I was able to get 2.292 (so, as you may have guessed, \ exact 2 was not the correct result either)." }], "Text", CellChangeTimes->{{3.454767906332025*^9, 3.4547679449884577`*^9}, { 3.454768260447653*^9, 3.4547683112005863`*^9}}], Cell[TextData[{ Cell[BoxData[ FormBox[ RowBox[{"j", " "}], TraditionalForm]], FormatType->"TraditionalForm"], ButtonBox["\[FilledLeftTriangle]\[ThickSpace]\[ThickSpace]\[ThickSpace]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPagePrevious"]}], ButtonNote->FEPrivate`FrontEndResource[ "FEStrings", "SlideshowPrevSlideText"]], "\[ThickSpace]\[ThickSpace]|\[ThickSpace]\[ThickSpace]", ButtonBox["\[ThickSpace]\[ThickSpace]\[ThickSpace]\[FilledRightTriangle]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageNext"]}], ButtonNote->FEPrivate`FrontEndResource[ "FEStrings", "SlideshowNextSlideText"]] }], "PreviousNext", CellChangeTimes->{{3.454768279233904*^9, 3.4547682859564133`*^9}}] }, Open ]], Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Troublesome aspects", "Section", CellChangeTimes->{{3.424789104977*^9, 3.424789114292745*^9}, 3.424789299633628*^9, {3.42479274300571*^9, 3.424792772016237*^9}, { 3.424792855532515*^9, 3.424792857438486*^9}, {3.424792942847715*^9, 3.424792946783374*^9}, {3.425078775130708*^9, 3.425078776673465*^9}, { 3.425132282397209*^9, 3.425132316907309*^9}, {3.425134477560227*^9, 3.425134491746237*^9}, {3.425135244600003*^9, 3.425135252263524*^9}, 3.42609080942988*^9, {3.454768169487207*^9, 3.454768175358253*^9}}], Cell["There are serious problems with this approach.", "Text", CellChangeTimes->{{3.454767967548568*^9, 3.454767978860735*^9}}], Cell["Difficult to get good precision", "Item", CellChangeTimes->{{3.4547678325796337`*^9, 3.454767883019609*^9}, { 3.454767985245481*^9, 3.454767993350169*^9}}], Cell["\<\ (Related, but more serious) It is quite difficult to bound the error. Indeed, \ it is not easy to show we have convergence.\ \>", "Item", CellChangeTimes->{{3.454767994172825*^9, 3.454768039638667*^9}, { 3.454768518994351*^9, 3.454768521249159*^9}}], Cell[TextData[{ ButtonBox["\[FilledLeftTriangle]\[ThickSpace]\[ThickSpace]\[ThickSpace]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPagePrevious"]}], ButtonNote->FEPrivate`FrontEndResource[ "FEStrings", "SlideshowPrevSlideText"]], "\[ThickSpace]\[ThickSpace]|\[ThickSpace]\[ThickSpace]", ButtonBox["\[ThickSpace]\[ThickSpace]\[ThickSpace]\[FilledRightTriangle]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageNext"]}], ButtonNote->FEPrivate`FrontEndResource[ "FEStrings", "SlideshowNextSlideText"]] }], "PreviousNext"] }, Open ]], Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Tischer's idea", "Section", CellChangeTimes->{{3.424789104977*^9, 3.424789114292745*^9}, 3.424789299633628*^9, {3.42479274300571*^9, 3.424792772016237*^9}, { 3.424792855532515*^9, 3.424792857438486*^9}, {3.424792942847715*^9, 3.424792946783374*^9}, {3.425078775130708*^9, 3.425078776673465*^9}, { 3.425132282397209*^9, 3.425132316907309*^9}, {3.425133267637193*^9, 3.425133291279485*^9}, {3.425135559500701*^9, 3.425135561739701*^9}, 3.4260908166502624`*^9, {3.4547683697945423`*^9, 3.454768372479816*^9}}], Cell["\<\ Figure out exact forms for some of the infinite sums, so as to avoid \ truncation. In parts we cannot compute exactly, show that error is much \ better than what we have from above approach.\ \>", "Text", CellChangeTimes->{{3.425135579726404*^9, 3.425135611870911*^9}, { 3.425135653301516*^9, 3.425135830505948*^9}, {3.425169108567298*^9, 3.425169134649443*^9}, {3.454768465939146*^9, 3.454768510898205*^9}, 3.4547685533945007`*^9, {3.454768607556073*^9, 3.4547686221134977`*^9}, { 3.4547700481905813`*^9, 3.454770068653493*^9}, 3.4553896893432407`*^9}] }, Open ]], Cell[TextData[{ "Start by writing ", Cell[BoxData[ FormBox[ RowBox[{"log", RowBox[{"(", RowBox[{"p", RowBox[{"(", "x", ")"}]}], ")"}]}], TraditionalForm]], FormatType->"TraditionalForm"], " (after a bit of algebra) as" }], "Text", CellChangeTimes->{{3.425135579726404*^9, 3.425135611870911*^9}, { 3.425135653301516*^9, 3.425135830505948*^9}, {3.425169108567298*^9, 3.425169134649443*^9}, {3.454768465939146*^9, 3.454768510898205*^9}, 3.4547685533945007`*^9, {3.454768607556073*^9, 3.454768630195046*^9}, { 3.454769010981347*^9, 3.4547690253818197`*^9}}], Cell[BoxData[ RowBox[{"x", "+", FractionBox[ SuperscriptBox["x", "2"], "2"], "+", RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"k", "=", "2"}], "\[Infinity]"], RowBox[{"(", RowBox[{ FractionBox[ SuperscriptBox["x", RowBox[{"k", "+", "1"}]], RowBox[{"k", "+", "1"}]], "-", FractionBox[ SuperscriptBox["x", RowBox[{"m", "[", "k", "]"}]], RowBox[{"k", "+", "1"}]]}], ")"}]}], " ", "-", RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"k", "=", "2"}], "\[Infinity]"], RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"j", "=", "2"}], "\[Infinity]"], RowBox[{ RowBox[{"(", FractionBox["1", "j"], ")"}], SuperscriptBox[ RowBox[{"(", FractionBox[ SuperscriptBox["x", RowBox[{"m", "[", "k", "]"}]], RowBox[{"k", "+", "1"}]], ")"}], "j"]}]}]}]}]], "DisplayFormula", CellChangeTimes->{{3.454768557218684*^9, 3.4547685872416973`*^9}}], Cell[TextData[{ "Proposition: This approaches ", Cell[BoxData[ FormBox[ RowBox[{"\[Gamma]", "+", RowBox[{"log", "(", "2", ")"}], "+", RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"k", "=", "2"}], "\[Infinity]"], RowBox[{"(", RowBox[{ FractionBox[ SuperscriptBox["x", RowBox[{"k", "+", "1"}]], RowBox[{"k", "+", "1"}]], "-", FractionBox[ SuperscriptBox["x", RowBox[{"m", "(", "k", ")"}]], RowBox[{"k", "+", "1"}]]}], ")"}]}]}], TraditionalForm]], "DisplayFormulaInline"], " as ", Cell[BoxData[ FormBox[ RowBox[{"x", "\[Rule]", "1"}], TraditionalForm]], "DisplayFormulaInline"], " from below." }], "Text", CellChangeTimes->{{3.4547690314336157`*^9, 3.4547690621924553`*^9}, { 3.4547690998786907`*^9, 3.454769161637848*^9}}], Cell[TextData[{ ButtonBox["\[FilledLeftTriangle]\[ThickSpace]\[ThickSpace]\[ThickSpace]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPagePrevious"]}], ButtonNote->FEPrivate`FrontEndResource[ "FEStrings", "SlideshowPrevSlideText"]], "\[ThickSpace]\[ThickSpace]|\[ThickSpace]\[ThickSpace]", ButtonBox["\[ThickSpace]\[ThickSpace]\[ThickSpace]\[FilledRightTriangle]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageNext"]}], ButtonNote->FEPrivate`FrontEndResource[ "FEStrings", "SlideshowNextSlideText"]] }], "PreviousNext"], Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Sketch of proof", "Section", CellChangeTimes->{{3.424789104977*^9, 3.424789114292745*^9}, 3.424789299633628*^9, {3.42479274300571*^9, 3.424792772016237*^9}, { 3.424792855532515*^9, 3.424792857438486*^9}, {3.424792942847715*^9, 3.424792946783374*^9}, {3.425078775130708*^9, 3.425078776673465*^9}, { 3.425132282397209*^9, 3.425132316907309*^9}, {3.425133267637193*^9, 3.425133291279485*^9}, {3.425135559500701*^9, 3.425135561739701*^9}, { 3.425169174839156*^9, 3.425169197959293*^9}, {3.425169254251434*^9, 3.425169256377747*^9}, 3.4260908193140926`*^9, {3.4547694149699183`*^9, 3.454769417000166*^9}, {3.4547695942673283`*^9, 3.4547695954495792`*^9}}], Cell["\<\ Clearly we only need focus on the double summation. 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We split by smallest divisors.\ \>", "Text", CellChangeTimes->{{3.425135579726404*^9, 3.425135611870911*^9}, { 3.425135653301516*^9, 3.425135830505948*^9}, 3.425169217458804*^9, { 3.425169615644165*^9, 3.4251696327484*^9}, {3.425170412198279*^9, 3.425170449094912*^9}, {3.454769931693038*^9, 3.454769940988326*^9}, { 3.454769997165187*^9, 3.454770007500832*^9}}], Cell[BoxData[ StyleBox[ RowBox[{" ", RowBox[{ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"k", "=", "2"}], "\[Infinity]"], RowBox[{"(", RowBox[{ FractionBox[ SuperscriptBox["x", RowBox[{"k", "+", "1"}]], RowBox[{"k", "+", "1"}]], "-", FractionBox[ SuperscriptBox["x", RowBox[{"m", "[", "k", "]"}]], RowBox[{"k", "+", "1"}]]}], ")"}]}], "=", RowBox[{ RowBox[{ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{ RowBox[{"d", "[", "k", "]"}], "=", "2"}], "\[Infinity]"], RowBox[{"(", RowBox[{ FractionBox[ SuperscriptBox["x", RowBox[{"k", "+", "1"}]], RowBox[{"k", "+", "1"}]], "-", FractionBox[ SuperscriptBox["x", RowBox[{"(", 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Strange...\ \>", "Text", CellChangeTimes->{{3.425172562558945*^9, 3.425172612163402*^9}, { 3.454786156063895*^9, 3.4547861709275703`*^9}, {3.454786349954176*^9, 3.454786352208902*^9}}] }, Open ]], Cell[TextData[{ ButtonBox["\[FilledLeftTriangle]\[ThickSpace]\[ThickSpace]\[ThickSpace]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPagePrevious"]}], ButtonNote->FEPrivate`FrontEndResource[ "FEStrings", "SlideshowPrevSlideText"]], "\[ThickSpace]\[ThickSpace]|\[ThickSpace]\[ThickSpace]", ButtonBox["\[ThickSpace]\[ThickSpace]\[ThickSpace]\[FilledRightTriangle]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageNext"]}], ButtonNote->FEPrivate`FrontEndResource[ "FEStrings", "SlideshowNextSlideText"]] }], "PreviousNext"], Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Further items...", "Section", CellChangeTimes->{{3.424789104977*^9, 3.424789114292745*^9}, 3.424789299633628*^9, {3.42479274300571*^9, 3.424792772016237*^9}, { 3.424792855532515*^9, 3.424792857438486*^9}, {3.424792942847715*^9, 3.424792946783374*^9}, {3.425078775130708*^9, 3.425078776673465*^9}, { 3.425132282397209*^9, 3.425132316907309*^9}, {3.425133267637193*^9, 3.425133291279485*^9}, {3.425135559500701*^9, 3.425135561739701*^9}, { 3.425169174839156*^9, 3.425169197959293*^9}, {3.425170967549881*^9, 3.425170981415762*^9}, {3.425171330929819*^9, 3.425171336323723*^9}, { 3.425172439732351*^9, 3.425172440947877*^9}, 3.4260908441197615`*^9, { 3.454786358209465*^9, 3.454786361728969*^9}}], Cell[TextData[{ "But stranger is the magnitudes of these coefficients. They are not even \ bounded by O", Cell[BoxData[ FormBox[ RowBox[{"(", FractionBox["1", "n"], ")"}], TraditionalForm]], "DisplayFormulaInline"], "." }], "Text", CellChangeTimes->{{3.425171349003178*^9, 3.425171374645817*^9}, { 3.425171407449081*^9, 3.425171412148048*^9}, {3.425171885997065*^9, 3.425171926601995*^9}, {3.425172449321006*^9, 3.425172513439877*^9}, { 3.454786364897661*^9, 3.454786420978857*^9}}], Cell["Can show:", "Text", CellChangeTimes->{{3.425172528974481*^9, 3.425172555476554*^9}, { 3.45478642835391*^9, 3.4547864505787067`*^9}, {3.4547864953975153`*^9, 3.4547865004338923`*^9}}], Cell[TextData[{ "They are bounded by O", Cell[BoxData[ FormBox[ RowBox[{"(", FractionBox[ RowBox[{"log", " ", "[", RowBox[{"log", "[", " ", "n", "]"}], "]"}], "n"], ")"}], TraditionalForm]], "DisplayFormulaInline"] }], "Item", CellChangeTimes->{{3.454786524371357*^9, 3.45478653139543*^9}}], Cell["\<\ This bound is tight (we can show that infinitely many coefficients will \ approach it closely).\ \>", "Item", CellChangeTimes->{{3.45478653326646*^9, 3.45478656979513*^9}, { 3.454786864287258*^9, 3.45478686429387*^9}}], Cell["\<\ The ones that approach closely have interesting factorization patterns (which \ is why they approach closely).\ \>", "Item", CellChangeTimes->{{3.454786865909067*^9, 3.45478688702951*^9}, { 3.454786961606411*^9, 3.454786987416629*^9}}], Cell["\<\ Figuring this out was more math than computation (our jobs are not going to \ the machines just yet).\ \>", "Item", CellChangeTimes->{{3.454786990726389*^9, 3.4547870537674217`*^9}}], Cell["\<\ Upshot: a naive summation will clearly give very poor convergence.\ \>", "Text", CellChangeTimes->{{3.45478645459389*^9, 3.454786468849959*^9}, { 3.454786576915618*^9, 3.454786578354764*^9}}] }, Open ]], Cell[TextData[{ ButtonBox["\[FilledLeftTriangle]\[ThickSpace]\[ThickSpace]\[ThickSpace]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPagePrevious"]}], ButtonNote->FEPrivate`FrontEndResource[ "FEStrings", "SlideshowPrevSlideText"]], "\[ThickSpace]\[ThickSpace]|\[ThickSpace]\[ThickSpace]", ButtonBox["\[ThickSpace]\[ThickSpace]\[ThickSpace]\[FilledRightTriangle]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageNext"]}], ButtonNote->FEPrivate`FrontEndResource[ "FEStrings", "SlideshowNextSlideText"]] }], "PreviousNext"], Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Some open problems", "Section", CellChangeTimes->{{3.424789104977*^9, 3.424789114292745*^9}, 3.424789299633628*^9, {3.42479274300571*^9, 3.424792772016237*^9}, { 3.424792855532515*^9, 3.424792857438486*^9}, {3.424792942847715*^9, 3.424792946783374*^9}, {3.425078775130708*^9, 3.425078776673465*^9}, { 3.425132282397209*^9, 3.425132316907309*^9}, {3.425133267637193*^9, 3.425133291279485*^9}, {3.425135559500701*^9, 3.425135561739701*^9}, { 3.425169174839156*^9, 3.425169197959293*^9}, {3.425170967549881*^9, 3.425170981415762*^9}, {3.425171330929819*^9, 3.425171336323723*^9}, { 3.425172439732351*^9, 3.425172440947877*^9}, 3.4260908441197615`*^9, { 3.454785904062817*^9, 3.45478591058952*^9}, {3.454786728436347*^9, 3.454786729716*^9}}], Cell["\<\ Understand the sign patterns of error approximants.\ \>", "Item", CellChangeTimes->{{3.425172562558945*^9, 3.425172612163402*^9}, { 3.454785945909033*^9, 3.454785948606694*^9}}], Cell["\<\ Find a more efficient way to compute the estimate to high precision.\ \>", "Item", CellChangeTimes->{{3.425172562558945*^9, 3.425172612163402*^9}, { 3.454785945909033*^9, 3.454785990190248*^9}}], Cell["Find an exact closed form for the limit.", "Item", CellChangeTimes->{{3.425172621025455*^9, 3.425172698536284*^9}, { 3.454785951806016*^9, 3.454785954227859*^9}, {3.4547862831684017`*^9, 3.454786284753997*^9}}] }, Open ]], Cell["\<\ Automate more of the symbolic analysis: some still requires manual \ intervention.\ \>", "Item", CellChangeTimes->{{3.425172621025455*^9, 3.425172698536284*^9}, { 3.454785951806016*^9, 3.454785954227859*^9}, {3.4547862831684017`*^9, 3.454786310384652*^9}, {3.454786740564958*^9, 3.454786754788183*^9}, { 3.454788389236672*^9, 3.454788389244235*^9}}], Cell["\<\ Determine whether the error bound/estimate is tight. 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Knopfmacher (1999a). Usenet news group post to \ comp.soft\[Hyphen]sys.math.mathematica. http://library.wolfram.com/mathgroup/archive/1999/Jan/msg00023.html\ \>", "Reference", CellChangeTimes->{{3.454785525032305*^9, 3.454785552198613*^9}}], Cell[TextData[{ "A. Knopfmacher and R. Warlimont (1995). Distinct degree factorizations for \ polynomials over a finite field. Transactions of the American Mathematical \ Society ", StyleBox["347", FontWeight->"Bold"], "(6):2235\[Hyphen]2243." }], "Reference", CellChangeTimes->{{3.454785525032305*^9, 3.454785552198613*^9}, { 3.454785584195799*^9, 3.454785607693672*^9}}], Cell["\<\ DL (1999). The evaluation of Knopfmacher's curious limit. Manuscript.\ \>", "Reference", CellChangeTimes->{{3.454785525032305*^9, 3.454785564891061*^9}, { 3.4547856178996267`*^9, 3.454785687388055*^9}}], Cell["\<\ J. Tischer (1999). Usenet news group post to \ comp.soft\[Hyphen]sys.math.mathematica. http://library.wolfram.com/mathgroup/archive/1999/Jan/msg00355.html\ \>", "Reference", CellChangeTimes->{{3.454785525032305*^9, 3.454785564891061*^9}, { 3.4547856178996267`*^9, 3.454785646447871*^9}}], Cell[TextData[{ ButtonBox["\[FilledLeftTriangle]\[ThickSpace]\[ThickSpace]\[ThickSpace]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPagePrevious"]}], ButtonNote->FEPrivate`FrontEndResource[ "FEStrings", "SlideshowPrevSlideText"]], "\[ThickSpace]\[ThickSpace]|\[ThickSpace]\[ThickSpace]", ButtonBox["\[ThickSpace]\[ThickSpace]\[ThickSpace]\[FilledRightTriangle]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageNext"]}], ButtonNote->FEPrivate`FrontEndResource[ "FEStrings", "SlideshowNextSlideText"]] }], 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