DCExamples.nb

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Cell["DirichletCharacters` Examples", "Title",
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Cell["The package, an overview", "Subchapter",
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Place the package and this notebook in the same directory, or place the \
package in $UserBaseDirectory<>\[CloseCurlyDoubleQuote]Applications\
\[CloseCurlyDoubleQuote]. The next command will then load the package.\
\>", "Text",
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You can view a list of all of the new functions, and particular information \
about each, as follows. Note that they are all in the context \
\[OpenCurlyDoubleQuote]DirichletCharacters`\[CloseCurlyDoubleQuote] (that is \
a back apostrophe!).\
\>", "Text",
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A standard thing people like to look at is a character table. Here\
\[CloseCurlyQuote]s the Character table for the 8 characters with modulus 20. \
Two things are noteworthy: (1) The characters are identified by Conrey\
\[CloseCurlyQuote]s notation, and (2) the table is symmetric due to Conrey\
\[CloseCurlyQuote]s numbering system.\
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 "Observe the notation for the characters: DC[modulus, index], where index \
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in the (undocumented!) Mathematica order. Note the lack of transpositional \
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Cell["Even more economical is a list of generators.", "Text",
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Cell["\<\
That is, the 80 elements in the group of characters modulo 200 can be built \
up from DC[200,101], DC[200,151], and DC[200,171] by multiplication. The \
package overloads the Times and Power functions.\
\>", "Text",
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Cell["\<\
The package provides the following test functions:
\tTrivialCharacterQ (has modulus 1?)
\tPrincipalCharacterQ (only takes values 0 and 1?)
\tRealCharacterQ (only takes real values?)
\tPrimitiveCharacterQ (not induced by a character with a smaller modulus?)\
\>", "Text",
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Cell["\<\
Additionally, the package overloads the following commands:
\tSign (\[Chi](-1), which is either 1 or -1)
\tEvenQ (is Sign 1)
\tOddQ (is Sign -1)\
\>", "Text",
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The character DC[200,171] is primitive: it is not induced by any character \
with smaller modulus:\
\>", "Text",
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The conductor of a character is the smallest possible modulus of an inducing \
character.\
\>", "Text",
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That is, chi and DC[2200,969] agree on inputs that are relatively prime to \
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\>", "Text",
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