Skip to content
This repository has been archived by the owner on Mar 17, 2023. It is now read-only.

Commit

Permalink
Updating the Mathematica supplementary notebook sources
Browse files Browse the repository at this point in the history
  • Loading branch information
maxieds committed Aug 17, 2020
1 parent 3e9a2ed commit 1e39005
Show file tree
Hide file tree
Showing 2 changed files with 297 additions and 19 deletions.
10 changes: 5 additions & 5 deletions MathematicaNotebooks/HatPiSummatoryFunctionsLowerBoundsProof.nb
Original file line number Diff line number Diff line change
Expand Up @@ -10,10 +10,10 @@
NotebookFileLineBreakTest
NotebookFileLineBreakTest
NotebookDataPosition[ 158, 7]
NotebookDataLength[ 17895400, 402782]
NotebookDataLength[ 17895403, 402782]
NotebookOptionsPosition[ 17892815, 402700]
NotebookOutlinePosition[ 17893153, 402715]
CellTagsIndexPosition[ 17893110, 402712]
NotebookOutlinePosition[ 17893156, 402715]
CellTagsIndexPosition[ 17893113, 402712]
WindowFrame->Normal*)

(* Beginning of Notebook Content *)
Expand Down Expand Up @@ -402699,7 +402699,7 @@ Cell[BoxData[
}, Open ]]
},
WindowSize->{1296, 704},
WindowMargins->{{0, Automatic}, {0, Automatic}},
WindowMargins->{{-69, Automatic}, {Automatic, 11}},
FrontEndVersion->"11.0 for Linux x86 (64-bit) (September 21, 2016)",
StyleDefinitions->"Default.nb"
]
Expand Down Expand Up @@ -402728,7 +402728,7 @@ Cell[12254, 284, 723, 13, 58, "Subtitle"],
Cell[12980, 299, 7456, 128, 78, "Input"],
Cell[CellGroupData[{
Cell[20461, 431, 2308, 57, 124, "Input"],
Cell[22772, 490, 17188, 501, 235, "Output"]
Cell[22772, 490, 17188, 501, 164, "Output"]
}, Open ]],
Cell[CellGroupData[{
Cell[39997, 996, 1928, 51, 101, "Input"],
Expand Down
306 changes: 292 additions & 14 deletions MathematicaNotebooks/LowerBoundsOnTheMertensFunctionProofs.nb
Original file line number Diff line number Diff line change
Expand Up @@ -10,10 +10,10 @@
NotebookFileLineBreakTest
NotebookFileLineBreakTest
NotebookDataPosition[ 158, 7]
NotebookDataLength[ 48134, 1285]
NotebookOptionsPosition[ 46016, 1211]
NotebookOutlinePosition[ 46357, 1226]
CellTagsIndexPosition[ 46314, 1223]
NotebookDataLength[ 55926, 1563]
NotebookOptionsPosition[ 53258, 1471]
NotebookOutlinePosition[ 53599, 1486]
CellTagsIndexPosition[ 53556, 1483]
WindowFrame->Normal*)

(* Beginning of Notebook Content *)
Expand Down Expand Up @@ -978,7 +978,7 @@ Cell[BoxData[
SqrtBox["2"], " ",
SqrtBox["t"]}]]], "Output",
CellChangeTimes->{3.8042980392775803`*^9, 3.805668982113467*^9}]
}, Open ]],
}, Closed]],

Cell[CellGroupData[{

Expand Down Expand Up @@ -1026,6 +1026,266 @@ Cell[BoxData[
RowBox[{"Log", "[",
RowBox[{"Log", "[", "x", "]"}], "]"}]]}]]], "Output",
CellChangeTimes->{3.804298051608395*^9}]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[{
RowBox[{
RowBox[{"derivTerms", "=",
RowBox[{
RowBox[{"1", "/",
RowBox[{"Sqrt", "[",
RowBox[{"Log", "[",
RowBox[{"Log", "[", "x", "]"}], "]"}], "]"}]}], "/",
RowBox[{"Power", "[",
RowBox[{"x", ",",
RowBox[{"1", "/", "4"}]}], "]"}]}]}], ";"}], "\[IndentingNewLine]",
RowBox[{
RowBox[{"remTerms", "=",
RowBox[{"Power", "[",
RowBox[{
RowBox[{"(",
RowBox[{"2", "t"}], ")"}], ",", "1"}], "]"}]}],
";"}], "\[IndentingNewLine]",
RowBox[{
RowBox[{
RowBox[{
RowBox[{"derivTerms", "*", "remTerms"}], "/.",
RowBox[{"(",
RowBox[{"x", "->",
RowBox[{"Exp", "[",
RowBox[{"Exp", "[",
RowBox[{"2", "t"}], "]"}], "]"}]}], ")"}]}], " ", "//", " ",
"FullSimplify"}], " ", "//", " ", "PowerExpand"}]}], "Input",
CellChangeTimes->{
3.804298038587513*^9, {3.8056689428027554`*^9, 3.805668980985664*^9}, {
3.806104581086488*^9, 3.8061045828133383`*^9}}],

Cell[BoxData[
RowBox[{
SqrtBox["2"], " ",
SuperscriptBox["\[ExponentialE]",
RowBox[{"-",
FractionBox[
SuperscriptBox["\[ExponentialE]",
RowBox[{"2", " ", "t"}]], "4"]}]], " ",
SqrtBox["t"]}]], "Output",
CellChangeTimes->{3.806104583851025*^9}]
}, Open ]],

Cell[CellGroupData[{

Cell[BoxData[{
RowBox[{
RowBox[{
RowBox[{"Integrate", "[",
RowBox[{
SuperscriptBox["\[ExponentialE]",
RowBox[{"-",
FractionBox[
SuperscriptBox["\[ExponentialE]",
RowBox[{"2", " ", "t"}]], "4"]}]], ",", "t"}], "]"}],
RowBox[{"(",
RowBox[{
SqrtBox["2"], " ",
SqrtBox["t"]}], ")"}]}], "/.",
RowBox[{"(",
RowBox[{"t", "\[Rule]",
RowBox[{
RowBox[{"Log", "[",
RowBox[{"Log", "[", "x", "]"}], "]"}], "/", "2"}]}],
")"}]}], "\[IndentingNewLine]",
RowBox[{
RowBox[{
RowBox[{"Integrate", "[",
RowBox[{
SuperscriptBox["\[ExponentialE]",
RowBox[{"-",
FractionBox[
SuperscriptBox["\[ExponentialE]",
RowBox[{"2", " ", "t"}]], "4"]}]], ",", "t"}], "]"}],
RowBox[{"(",
RowBox[{
SqrtBox["2"], " ",
SqrtBox["t"]}], ")"}]}], "/.",
RowBox[{"(",
RowBox[{"t", "\[Rule]",
RowBox[{
RowBox[{"Log", "[",
RowBox[{"Log", "[", "x", "]"}], "]"}], "/", "4"}]}], ")"}]}]}], "Input",\

CellChangeTimes->{
3.80429804995004*^9, {3.806104616532288*^9, 3.806104619179631*^9}, {
3.806104705620675*^9, 3.806104707243229*^9}}],

Cell[BoxData[
RowBox[{
FractionBox["1", "2"], " ",
RowBox[{"ExpIntegralEi", "[",
RowBox[{"-",
FractionBox[
RowBox[{"Log", "[", "x", "]"}], "4"]}], "]"}], " ",
SqrtBox[
RowBox[{"Log", "[",
RowBox[{"Log", "[", "x", "]"}], "]"}]]}]], "Output",
CellChangeTimes->{3.806104619789311*^9, 3.806104708110055*^9}],

Cell[BoxData[
FractionBox[
RowBox[{
RowBox[{"ExpIntegralEi", "[",
RowBox[{
RowBox[{"-",
FractionBox["1", "4"]}], " ",
SqrtBox[
RowBox[{"Log", "[", "x", "]"}]]}], "]"}], " ",
SqrtBox[
RowBox[{"Log", "[",
RowBox[{"Log", "[", "x", "]"}], "]"}]]}],
RowBox[{"2", " ",
SqrtBox["2"]}]]], "Output",
CellChangeTimes->{3.806104619789311*^9, 3.806104708111537*^9}]
}, Open ]],

Cell[CellGroupData[{

Cell[BoxData[{
RowBox[{
RowBox[{"derivTerms", "=",
RowBox[{
RowBox[{"1", "/",
RowBox[{"Sqrt", "[",
RowBox[{"Log", "[",
RowBox[{"Log", "[", "x", "]"}], "]"}], "]"}]}], "/",
RowBox[{"Power", "[",
RowBox[{"x", ",",
RowBox[{"1", "/", "4"}]}], "]"}]}]}], ";"}], "\[IndentingNewLine]",
RowBox[{
RowBox[{"remTerms", "=",
RowBox[{"Power", "[",
RowBox[{
RowBox[{"(",
RowBox[{
RowBox[{"2", "t"}], "-", "1"}], ")"}], ",", "1"}], "]"}]}],
";"}], "\[IndentingNewLine]",
RowBox[{
RowBox[{
RowBox[{
RowBox[{"derivTerms", "*", "remTerms"}], "/.",
RowBox[{"(",
RowBox[{"x", "->",
RowBox[{"Exp", "[",
RowBox[{"Exp", "[",
RowBox[{
RowBox[{"2", "t"}], "-", "1"}], "]"}], "]"}]}], ")"}]}], " ", "//",
" ", "FullSimplify"}], " ", "//", " ", "PowerExpand"}]}], "Input",
CellChangeTimes->{
3.804298038587513*^9, {3.8056689428027554`*^9, 3.805668980985664*^9}, {
3.806104581086488*^9, 3.8061045828133383`*^9}, {3.806104837995153*^9,
3.8061048432277822`*^9}}],

Cell[BoxData[
RowBox[{
SuperscriptBox["\[ExponentialE]",
RowBox[{
RowBox[{"-",
FractionBox["1", "4"]}], " ",
SuperscriptBox["\[ExponentialE]",
RowBox[{
RowBox[{"-", "1"}], "+",
RowBox[{"2", " ", "t"}]}]]}]], " ",
SqrtBox[
RowBox[{
RowBox[{"-", "1"}], "+",
RowBox[{"2", " ", "t"}]}]]}]], "Output",
CellChangeTimes->{{3.806104839334383*^9, 3.8061048461830797`*^9}}]
}, Open ]],

Cell[CellGroupData[{

Cell[BoxData[{
RowBox[{
RowBox[{
RowBox[{"Integrate", "[",
RowBox[{
SuperscriptBox["\[ExponentialE]",
RowBox[{"-",
FractionBox[
SuperscriptBox["\[ExponentialE]",
RowBox[{
RowBox[{"2", " ", "t"}], "-", "1"}]], "4"]}]], ",", "t"}], "]"}],
RowBox[{"(", " ",
SqrtBox[
RowBox[{
RowBox[{"-", "1"}], "+",
RowBox[{"2", " ", "t"}]}]], ")"}]}], "/.",
RowBox[{"(",
RowBox[{"t", "\[Rule]",
RowBox[{
RowBox[{"Log", "[",
RowBox[{"Log", "[", "x", "]"}], "]"}], "/", "2"}]}],
")"}]}], "\[IndentingNewLine]",
RowBox[{
RowBox[{
RowBox[{"Integrate", "[",
RowBox[{
SuperscriptBox["\[ExponentialE]",
RowBox[{"-",
FractionBox[
SuperscriptBox["\[ExponentialE]",
RowBox[{
RowBox[{"2", " ", "t"}], "-", "1"}]], "4"]}]], ",", "t"}], "]"}],
RowBox[{"(", " ",
SqrtBox[
RowBox[{
RowBox[{"-", "1"}], "+",
RowBox[{"2", " ", "t"}]}]], ")"}]}], "/.",
RowBox[{"(",
RowBox[{"t", "\[Rule]",
RowBox[{
RowBox[{"Log", "[",
RowBox[{"Log", "[", "x", "]"}], "]"}], "/", "4"}]}], ")"}]}]}], "Input",\

CellChangeTimes->{
3.80429804995004*^9, {3.806104616532288*^9, 3.806104619179631*^9}, {
3.806104705620675*^9, 3.806104707243229*^9}, {3.806104856535502*^9,
3.8061048721962223`*^9}, {3.806104974727028*^9, 3.8061049799150143`*^9}}],

Cell[BoxData[
RowBox[{
FractionBox["1", "2"], " ",
RowBox[{"ExpIntegralEi", "[",
RowBox[{"-",
FractionBox[
RowBox[{"Log", "[", "x", "]"}],
RowBox[{"4", " ", "\[ExponentialE]"}]]}], "]"}], " ",
SqrtBox[
RowBox[{
RowBox[{"-", "1"}], "+",
RowBox[{"Log", "[",
RowBox[{"Log", "[", "x", "]"}], "]"}]}]]}]], "Output",
CellChangeTimes->{
3.806104873400137*^9, {3.806104975458152*^9, 3.8061049802453012`*^9}}],

Cell[BoxData[
RowBox[{
FractionBox["1", "2"], " ",
RowBox[{"ExpIntegralEi", "[",
RowBox[{"-",
FractionBox[
SqrtBox[
RowBox[{"Log", "[", "x", "]"}]],
RowBox[{"4", " ", "\[ExponentialE]"}]]}], "]"}], " ",
SqrtBox[
RowBox[{
RowBox[{"-", "1"}], "+",
RowBox[{
FractionBox["1", "2"], " ",
RowBox[{"Log", "[",
RowBox[{"Log", "[", "x", "]"}], "]"}]}]}]]}]], "Output",
CellChangeTimes->{
3.806104873400137*^9, {3.806104975458152*^9, 3.806104980247033*^9}}]
}, Open ]],

Cell[CellGroupData[{
Expand Down Expand Up @@ -1269,23 +1529,41 @@ Cell[25795, 731, 11582, 195, 139, "Input"],
Cell[CellGroupData[{
Cell[37402, 930, 986, 31, 79, "Input"],
Cell[38391, 963, 474, 16, 68, "Output"]
}, Open ]],
}, Closed]],
Cell[CellGroupData[{
Cell[38902, 984, 686, 23, 64, "Input"],
Cell[38902, 984, 686, 23, 58, "Input"],
Cell[39591, 1009, 523, 18, 66, "Output"]
}, Closed]],
Cell[CellGroupData[{
Cell[40151, 1032, 977, 30, 73, "Input"],
Cell[41131, 1064, 274, 9, 44, "Output"]
}, Open ]],
Cell[CellGroupData[{
Cell[41442, 1078, 1185, 41, 96, "Input"],
Cell[42630, 1121, 335, 10, 49, "Output"],
Cell[42968, 1133, 410, 14, 65, "Output"]
}, Open ]],
Cell[CellGroupData[{
Cell[43415, 1152, 1087, 33, 79, "Input"],
Cell[44505, 1187, 420, 14, 40, "Output"]
}, Open ]],
Cell[CellGroupData[{
Cell[44962, 1206, 1423, 46, 96, "Input"],
Cell[46388, 1254, 450, 14, 50, "Output"],
Cell[46841, 1270, 515, 17, 92, "Output"]
}, Open ]],
Cell[CellGroupData[{
Cell[40151, 1032, 1766, 51, 123, "Input"],
Cell[41920, 1085, 366, 11, 40, "Output"],
Cell[42289, 1098, 325, 9, 40, "Output"],
Cell[42617, 1109, 640, 20, 40, "Output"]
Cell[47393, 1292, 1766, 51, 123, "Input"],
Cell[49162, 1345, 366, 11, 40, "Output"],
Cell[49531, 1358, 325, 9, 40, "Output"],
Cell[49859, 1369, 640, 20, 40, "Output"]
}, Open ]],
Cell[CellGroupData[{
Cell[43294, 1134, 1345, 35, 48, "Input"],
Cell[44642, 1171, 796, 24, 95, "Output"]
Cell[50536, 1394, 1345, 35, 48, "Input"],
Cell[51884, 1431, 796, 24, 62, "Output"]
}, Open ]]
}, Open ]],
Cell[45465, 1199, 535, 9, 58, "Subtitle"]
Cell[52707, 1459, 535, 9, 58, "Subtitle"]
}, Open ]]
}
]
Expand Down

0 comments on commit 1e39005

Please sign in to comment.