Dirichlet characters

Symbol: DirichletCharacter —

\chi_{q \, . \, \ell}

— Dirichlet character

DirichletCharacter(q, ell), rendered as

\chi_{q \, . \, \ell}

, represents the Dirichlet character with Conrey label

\left(q, \ell\right)

.

A character represents an object

\chi

that can be called (

\chi(n)

) as a function from

\mathbb{Z}

to

\mathbb{C}

.

DirichletCharacter(q, ell, n), rendered as

\chi_{q \, . \, \ell}\!\left(n\right)

, represents the Dirichlet character with Conrey label

\left(q, \ell\right)

evaluated at the integer

n

.

The Conrey label consists of integers

q \in \mathbb{Z}_{\ge 1}

and

\ell \in \{1, 2, \ldots, \max\!\left(q, 2\right) - 1\}

such that

\gcd\!\left(\ell, q\right) = 1

. In this scheme

\chi_{q \, . \, 1}

always represents the trivial/principal character (taking only values 0 and 1) modulo

q

. Non-principal characters are defined by 4cf4e4 when

q

is an odd prime power, by fc4f6a and 03fbe8 when

q

is an even prime power, and in general by factoring

q

into prime powers using 2a48bd.

References:

http://www.lmfdb.org/Character/Labels

Definitions:

Fungrim symbol	Notation	Short description
`DirichletCharacter`	$\chi_{q \, . \, \ell}$	Dirichlet character
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`Range`	$\{a, a + 1, \ldots, b\}$	Integers between given endpoints
`GCD`	$\gcd\!\left(a, b\right)$	Greatest common divisor

Source code for this entry:

Entry(ID("c7e2fb"),
    SymbolDefinition(DirichletCharacter, DirichletCharacter(q, ell), "Dirichlet character"),
    Description(SourceForm(DirichletCharacter(q, ell)), ", rendered as", DirichletCharacter(q, ell), ", represents the Dirichlet character with Conrey label", Tuple(q, ell), "."),
    Description("A character represents an object", chi, " that can be called (", chi(n), ") as a function from", ZZ, "to", CC, "."),
    Description(SourceForm(DirichletCharacter(q, ell, n)), ", rendered as", DirichletCharacter(q, ell, n), ", represents the Dirichlet character with Conrey label", Tuple(q, ell), "evaluated at the integer", n, "."),
    Description("The Conrey label consists of integers", Element(q, ZZGreaterEqual(1)), "and", Element(ell, Range(1, Sub(Max(q, 2), 1))), "such that", Equal(GCD(ell, q), 1), ". ", "In this scheme", DirichletCharacter(q, 1), "always represents the trivial/principal character (taking only values 0 and 1) modulo", q, ". ", "Non-principal characters are defined by", EntryReference("4cf4e4"), "when", q, "is an odd prime power, by", EntryReference("fc4f6a"), "and", EntryReference("03fbe8"), "when", q, "is an even prime power, and in general by factoring", q, "into prime powers using", EntryReference("2a48bd"), "."),
    References("http://www.lmfdb.org/Character/Labels"))

e810d8

Symbol: DirichletGroup —

G_{q}

— Dirichlet characters with given modulus

DirichletGroup(q), rendered as

G_{q}

, represents the set of Dirichlet characters modulo

q

, given

q \in \mathbb{Z}_{\ge 1}

.

Dirichlet characters can be defined axiomatically as functions from

\mathbb{Z}

to

\mathbb{C}

satisfying the properties in formulas 1c3957, 0851cf, and afd0c5.

In this definition, the modulus

q

is not an attribute of the character; for example the character giving the sequence

\left[0, 1, 0, 1, \ldots\right]

is an element of both

G_{2}

and

G_{4}

.

A more explicit construction of the characters is possible using the Conrey numbering scheme, which is implemented by DirichletCharacter.

Definitions:

Fungrim symbol	Notation	Short description
`DirichletGroup`	$G_{q}$	Dirichlet characters with given modulus
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers
`DirichletCharacter`	$\chi_{q \, . \, \ell}$	Dirichlet character

Source code for this entry:

Entry(ID("e810d8"),
    SymbolDefinition(DirichletGroup, DirichletGroup(q), "Dirichlet characters with given modulus"),
    Description(SourceForm(DirichletGroup(q)), ", rendered as", DirichletGroup(q), ", represents the set of Dirichlet characters modulo", q, ", given", Element(q, ZZGreaterEqual(1)), "."),
    Description("Dirichlet characters can be defined axiomatically as functions from", ZZ, "to", CC, "satisfying the properties in formulas", EntryReference("1c3957"), ", ", EntryReference("0851cf"), ", and", EntryReference("afd0c5"), "."),
    Description("In this definition, the modulus", q, "is not an attribute of the character; for example", "the character giving the sequence", List(0, 1, 0, 1, Ellipsis), "is an element of both", DirichletGroup(2), "and", DirichletGroup(4), "."),
    Description("A more explicit construction of the characters is possible using the Conrey numbering scheme, which is implemented by", SourceForm(DirichletCharacter), "."))

2f52bc

Symbol: PrimitiveDirichletCharacters —

G^{\text{Primitive}}_{q}

— Primitive Dirichlet characters with given modulus

PrimitiveDirichletCharacters(q), rendered as

G^{\text{Primitive}}_{q}

, represents the set of primitive Dirichlet characters modulo

q

, given

q \in \mathbb{Z}_{\ge 1}

. Primitive characters are defined in ed65c8.

Definitions:

Fungrim symbol	Notation	Short description
`PrimitiveDirichletCharacters`	$G^{\text{Primitive}}_{q}$	Primitive Dirichlet characters with given modulus
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("2f52bc"),
    SymbolDefinition(PrimitiveDirichletCharacters, PrimitiveDirichletCharacters(q), "Primitive Dirichlet characters with given modulus"),
    Description(SourceForm(PrimitiveDirichletCharacters(q)), ", rendered as", PrimitiveDirichletCharacters(q), ", represents the set of primitive Dirichlet characters modulo", q, ", given", Element(q, ZZGreaterEqual(1)), ". Primitive characters are defined in ", EntryReference("ed65c8"), "."))

d9a187

\chi(n) = \chi_{q \, . \, \ell}\!\left(n\right)\; \text{ where } \chi = \chi_{q \, . \, \ell}

This is simply a syntactical definition of character evaluation.

Assumptions:

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \ell \in \{1, 2, \ldots, \max\!\left(q, 2\right) - 1\} \;\mathbin{\operatorname{and}}\; \gcd\!\left(\ell, q\right) = 1 \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}

TeX:

\chi(n) = \chi_{q \, . \, \ell}\!\left(n\right)\; \text{ where } \chi = \chi_{q \, . \, \ell}

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \ell \in \{1, 2, \ldots, \max\!\left(q, 2\right) - 1\} \;\mathbin{\operatorname{and}}\; \gcd\!\left(\ell, q\right) = 1 \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}

Definitions:

Fungrim symbol	Notation	Short description
`DirichletCharacter`	$\chi_{q \, . \, \ell}$	Dirichlet character
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`Range`	$\{a, a + 1, \ldots, b\}$	Integers between given endpoints
`GCD`	$\gcd\!\left(a, b\right)$	Greatest common divisor
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("d9a187"),
    Formula(Where(Equal(chi(n), DirichletCharacter(q, ell, n)), Equal(chi, DirichletCharacter(q, ell)))),
    Description("This is simply a syntactical definition of character evaluation."),
    Variables(q, ell, n),
    Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(ell, Range(1, Sub(Max(q, 2), 1))), Equal(GCD(ell, q), 1), Element(n, ZZ))))

57d31a

\chi(n) \in \left\{ {e}^{2 \pi i k / r} : k \in \mathbb{Z} \right\} \cup \left\{0\right\}\; \text{ where } r = \varphi(q)

Assumptions:

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G_{q} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}

TeX:

\chi(n) \in \left\{ {e}^{2 \pi i k / r} : k \in \mathbb{Z} \right\} \cup \left\{0\right\}\; \text{ where } r = \varphi(q)

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G_{q} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}

Definitions:

Fungrim symbol	Notation	Short description
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`ZZ`	$\mathbb{Z}$	Integers
`Totient`	$\varphi(n)$	Euler totient function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`DirichletGroup`	$G_{q}$	Dirichlet characters with given modulus

Source code for this entry:

Entry(ID("57d31a"),
    Formula(Where(Element(chi(n), Union(Set(Exp(Div(Mul(Mul(Mul(2, Pi), ConstI), k), r)), ForElement(k, ZZ)), Set(0))), Equal(r, Totient(q)))),
    Variables(q, chi, n),
    Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(chi, DirichletGroup(q)), Element(n, ZZ))))

1c3957

\chi\!\left(n + q\right) = \chi(n)

Assumptions:

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G_{q} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}

TeX:

\chi\!\left(n + q\right) = \chi(n)

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G_{q} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}

Definitions:

Fungrim symbol	Notation	Short description
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`DirichletGroup`	$G_{q}$	Dirichlet characters with given modulus
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("1c3957"),
    Formula(Equal(chi(Add(n, q)), chi(n))),
    Variables(q, chi, n),
    Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(chi, DirichletGroup(q)), Element(n, ZZ))))

0851cf

\chi\!\left(m n\right) = \chi(m) \chi(n)

Assumptions:

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G_{q} \;\mathbin{\operatorname{and}}\; m \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}

TeX:

\chi\!\left(m n\right) = \chi(m) \chi(n)

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G_{q} \;\mathbin{\operatorname{and}}\; m \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}

Definitions:

Fungrim symbol	Notation	Short description
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`DirichletGroup`	$G_{q}$	Dirichlet characters with given modulus
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("0851cf"),
    Formula(Equal(chi(Mul(m, n)), Mul(chi(m), chi(n)))),
    Variables(q, chi, m, n),
    Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(chi, DirichletGroup(q)), Element(m, ZZ), Element(n, ZZ))))

afd0c5

\left(\chi(n) = 0\right) \iff \left(\gcd\!\left(n, q\right) \ne 1\right)

Assumptions:

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G_{q} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}

TeX:

\left(\chi(n) = 0\right) \iff \left(\gcd\!\left(n, q\right) \ne 1\right)

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G_{q} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}

Definitions:

Fungrim symbol	Notation	Short description
`GCD`	$\gcd\!\left(a, b\right)$	Greatest common divisor
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`DirichletGroup`	$G_{q}$	Dirichlet characters with given modulus
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("afd0c5"),
    Formula(Equivalent(Equal(chi(n), 0), NotEqual(GCD(n, q), 1))),
    Variables(q, chi, n),
    Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(chi, DirichletGroup(q)), Element(n, ZZ))))

bdf58d

\chi(1) = 1

Assumptions:

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G_{q}

TeX:

\chi(1) = 1

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G_{q}

Definitions:

Fungrim symbol	Notation	Short description
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`DirichletGroup`	$G_{q}$	Dirichlet characters with given modulus

Source code for this entry:

Entry(ID("bdf58d"),
    Formula(Equal(chi(1), 1)),
    Variables(q, chi),
    Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(chi, DirichletGroup(q)))))

d29554

\chi(-1) \in \left\{1, -1\right\}

Assumptions:

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G_{q}

TeX:

\chi(-1) \in \left\{1, -1\right\}

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G_{q}

Definitions:

Fungrim symbol	Notation	Short description
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`DirichletGroup`	$G_{q}$	Dirichlet characters with given modulus

Source code for this entry:

Entry(ID("d29554"),
    Formula(Element(chi(-1), Set(1, -1))),
    Variables(q, chi),
    Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(chi, DirichletGroup(q)))))

458198

\chi(0) = \begin{cases} 1, & q = 1\\0, & q \ne 1\\ \end{cases}

Assumptions:

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G_{q}

TeX:

\chi(0) = \begin{cases} 1, & q = 1\\0, & q \ne 1\\ \end{cases}

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G_{q}

Definitions:

Fungrim symbol	Notation	Short description
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`DirichletGroup`	$G_{q}$	Dirichlet characters with given modulus

Source code for this entry:

Entry(ID("458198"),
    Formula(Equal(chi(0), Cases(Tuple(1, Equal(q, 1)), Tuple(0, NotEqual(q, 1))))),
    Variables(q, chi),
    Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(chi, DirichletGroup(q)))))

d8c6d1

\chi_{q \, . \, 1}\!\left(n\right) = \begin{cases} 1, & \gcd\!\left(n, q\right) = 1\\0, & \text{otherwise}\\ \end{cases}

Assumptions:

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}

TeX:

\chi_{q \, . \, 1}\!\left(n\right) = \begin{cases} 1, & \gcd\!\left(n, q\right) = 1\\0, & \text{otherwise}\\ \end{cases}

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}

Definitions:

Fungrim symbol	Notation	Short description
`DirichletCharacter`	$\chi_{q \, . \, \ell}$	Dirichlet character
`GCD`	$\gcd\!\left(a, b\right)$	Greatest common divisor
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("d8c6d1"),
    Formula(Equal(DirichletCharacter(q, 1, n), Cases(Tuple(1, Equal(GCD(n, q), 1)), Tuple(0, Otherwise)))),
    Variables(q, n),
    Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(n, ZZ))))

47d430

G_{q} = \left\{ \chi_{q \, . \, \ell} : \ell \in \{1, 2, \ldots, \max\!\left(q, 2\right) - 1\} \;\mathbin{\operatorname{and}}\; \gcd\!\left(\ell, q\right) = 1 \right\}

Assumptions:

q \in \mathbb{Z}_{\ge 1}

TeX:

G_{q} = \left\{ \chi_{q \, . \, \ell} : \ell \in \{1, 2, \ldots, \max\!\left(q, 2\right) - 1\} \;\mathbin{\operatorname{and}}\; \gcd\!\left(\ell, q\right) = 1 \right\}

q \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`DirichletGroup`	$G_{q}$	Dirichlet characters with given modulus
`DirichletCharacter`	$\chi_{q \, . \, \ell}$	Dirichlet character
`Range`	$\{a, a + 1, \ldots, b\}$	Integers between given endpoints
`GCD`	$\gcd\!\left(a, b\right)$	Greatest common divisor
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("47d430"),
    Formula(Equal(DirichletGroup(q), Set(DirichletCharacter(q, ell), For(ell), And(Element(ell, Range(1, Sub(Max(q, 2), 1))), Equal(GCD(ell, q), 1))))),
    Variables(q),
    Assumptions(Element(q, ZZGreaterEqual(1))))

ed65c8

G^{\text{Primitive}}_{q} = \left\{ \chi : \chi \in G_{q} \;\mathbin{\operatorname{and}}\; \left[a \equiv 1 \pmod {d} \;\mathbin{\operatorname{and}}\; \gcd\!\left(a, q\right) = 1 \;\mathbin{\operatorname{and}}\; \chi(a) \ne 1 \;\text{ for some } a \in \{0, 1, \ldots, q - 1\} \;\text{ for all } d \in \{1, 2, \ldots, q - 1\} \text{ with } d \mid q\right] \right\}

Assumptions:

q \in \mathbb{Z}_{\ge 1}

References:

T. Apostol (1976), Introduction to Analytic Number Theory, Springer. Chapter 8.7.

TeX:

G^{\text{Primitive}}_{q} = \left\{ \chi : \chi \in G_{q} \;\mathbin{\operatorname{and}}\; \left[a \equiv 1 \pmod {d} \;\mathbin{\operatorname{and}}\; \gcd\!\left(a, q\right) = 1 \;\mathbin{\operatorname{and}}\; \chi(a) \ne 1 \;\text{ for some } a \in \{0, 1, \ldots, q - 1\} \;\text{ for all } d \in \{1, 2, \ldots, q - 1\} \text{ with } d \mid q\right] \right\}

q \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`PrimitiveDirichletCharacters`	$G^{\text{Primitive}}_{q}$	Primitive Dirichlet characters with given modulus
`DirichletGroup`	$G_{q}$	Dirichlet characters with given modulus
`GCD`	$\gcd\!\left(a, b\right)$	Greatest common divisor
`Range`	$\{a, a + 1, \ldots, b\}$	Integers between given endpoints
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("ed65c8"),
    Formula(Equal(PrimitiveDirichletCharacters(q), Set(chi, For(chi), And(Element(chi, DirichletGroup(q)), Brackets(All(Exists(And(CongruentMod(a, 1, d), Equal(GCD(a, q), 1), NotEqual(chi(a), 1)), ForElement(a, Range(0, Sub(q, 1)))), ForElement(d, Range(1, Sub(q, 1))), Divides(d, q))))))),
    Variables(q),
    Assumptions(Element(q, ZZGreaterEqual(1))),
    References("T. Apostol (1976), Introduction to Analytic Number Theory, Springer. Chapter 8.7."))

62f7d5

\# G_{q} = \varphi(q)

Assumptions:

q \in \mathbb{Z}_{\ge 1}

TeX:

\# G_{q} = \varphi(q)

q \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`Cardinality`	$\# S$	Set cardinality
`DirichletGroup`	$G_{q}$	Dirichlet characters with given modulus
`Totient`	$\varphi(n)$	Euler totient function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("62f7d5"),
    Formula(Equal(Cardinality(DirichletGroup(q)), Totient(q))),
    Variables(q),
    Assumptions(Element(q, ZZGreaterEqual(1))))

0ba38f

\# G^{\text{Primitive}}_{q} = \sum_{d \mid q} \varphi(d) \mu\!\left(\frac{q}{d}\right)

Assumptions:

q \in \mathbb{Z}_{\ge 1}

References:

http://oeis.org/A007431

TeX:

\# G^{\text{Primitive}}_{q} = \sum_{d \mid q} \varphi(d) \mu\!\left(\frac{q}{d}\right)

q \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`Cardinality`	$\# S$	Set cardinality
`PrimitiveDirichletCharacters`	$G^{\text{Primitive}}_{q}$	Primitive Dirichlet characters with given modulus
`DivisorSum`	$\sum_{k \mid n} f(k)$	Sum over divisors
`Totient`	$\varphi(n)$	Euler totient function
`MoebiusMu`	$\mu(n)$	Möbius function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("0ba38f"),
    Formula(Equal(Cardinality(PrimitiveDirichletCharacters(q)), DivisorSum(Mul(Totient(d), MoebiusMu(Div(q, d))), For(d, q)))),
    Variables(q),
    Assumptions(Element(q, ZZGreaterEqual(1))),
    References("http://oeis.org/A007431"))

f88596

\lim_{N \to \infty} \frac{\sum_{q=1}^{N} \# G_{q}}{\frac{1}{2} {N}^{2}} = \frac{6}{{\pi}^{2}}

TeX:

\lim_{N \to \infty} \frac{\sum_{q=1}^{N} \# G_{q}}{\frac{1}{2} {N}^{2}} = \frac{6}{{\pi}^{2}}

Definitions:

Fungrim symbol	Notation	Short description
`SequenceLimit`	$\lim_{n \to a} f(n)$	Limiting value of sequence
`Sum`	$\sum_{n} f(n)$	Sum
`Cardinality`	$\# S$	Set cardinality
`DirichletGroup`	$G_{q}$	Dirichlet characters with given modulus
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`Pi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("f88596"),
    Formula(Equal(SequenceLimit(Div(Sum(Cardinality(DirichletGroup(q)), For(q, 1, N)), Mul(Div(1, 2), Pow(N, 2))), For(N, Infinity)), Div(6, Pow(Pi, 2)))))

3b43b0

\lim_{N \to \infty} \frac{\sum_{q=1}^{N} \# G^{\text{Primitive}}_{q}}{\sum_{q=1}^{N} \# G_{q}} = \frac{6}{{\pi}^{2}}

References:

H. Jager, On the number of Dirichlet characters with modulus not exceeding x, Indagationes Mathematicae, Volume 76, Issue 5, 1973, Pages 452-455, https://doi.org/10.1016/1385-7258(73)90069-3

TeX:

\lim_{N \to \infty} \frac{\sum_{q=1}^{N} \# G^{\text{Primitive}}_{q}}{\sum_{q=1}^{N} \# G_{q}} = \frac{6}{{\pi}^{2}}

Definitions:

Fungrim symbol	Notation	Short description
`SequenceLimit`	$\lim_{n \to a} f(n)$	Limiting value of sequence
`Sum`	$\sum_{n} f(n)$	Sum
`Cardinality`	$\# S$	Set cardinality
`PrimitiveDirichletCharacters`	$G^{\text{Primitive}}_{q}$	Primitive Dirichlet characters with given modulus
`DirichletGroup`	$G_{q}$	Dirichlet characters with given modulus
`Infinity`	$\infty$	Positive infinity
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("3b43b0"),
    Formula(Equal(SequenceLimit(Div(Sum(Cardinality(PrimitiveDirichletCharacters(q)), For(q, 1, N)), Sum(Cardinality(DirichletGroup(q)), For(q, 1, N))), For(N, Infinity)), Div(6, Pow(Pi, 2)))),
    References("H. Jager, On the number of Dirichlet characters with modulus not exceeding x, Indagationes Mathematicae, Volume 76, Issue 5, 1973, Pages 452-455, https://doi.org/10.1016/1385-7258(73)90069-3"))

a7d592

\chi = {\chi}_{0} {\chi}_{1} \;\text{ for some } \left(d, {\chi}_{0}\right) \text{ with } d \in \{1, 2, \ldots, q\} \;\mathbin{\operatorname{and}}\; d \mid q \;\mathbin{\operatorname{and}}\; {\chi}_{0} \in G^{\text{Primitive}}_{d}\; \text{ where } {\chi}_{1} = \chi_{q \, . \, 1}

Assumptions:

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G_{q}

TeX:

\chi = {\chi}_{0} {\chi}_{1} \;\text{ for some } \left(d, {\chi}_{0}\right) \text{ with } d \in \{1, 2, \ldots, q\} \;\mathbin{\operatorname{and}}\; d \mid q \;\mathbin{\operatorname{and}}\; {\chi}_{0} \in G^{\text{Primitive}}_{d}\; \text{ where } {\chi}_{1} = \chi_{q \, . \, 1}

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G_{q}

Definitions:

Fungrim symbol	Notation	Short description
`Range`	$\{a, a + 1, \ldots, b\}$	Integers between given endpoints
`PrimitiveDirichletCharacters`	$G^{\text{Primitive}}_{q}$	Primitive Dirichlet characters with given modulus
`DirichletCharacter`	$\chi_{q \, . \, \ell}$	Dirichlet character
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`DirichletGroup`	$G_{q}$	Dirichlet characters with given modulus

Source code for this entry:

Entry(ID("a7d592"),
    Formula(Where(Exists(Equal(chi, Mul(Subscript(chi, 0), Subscript(chi, 1))), For(Tuple(d, Subscript(chi, 0))), And(Element(d, Range(1, q)), Divides(d, q), Element(Subscript(chi, 0), PrimitiveDirichletCharacters(d)))), Equal(Subscript(chi, 1), DirichletCharacter(q, 1)))),
    Variables(q, Subscript(chi, 0)),
    Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(chi, DirichletGroup(q)))))

2a48bd

\chi_{{q}_{1} {q}_{2} \, . \, \ell} = \chi_{{q}_{1} \, . \, \ell \bmod {q}_{1}} \chi_{{q}_{2} \, . \, \ell \bmod {q}_{2}}

Assumptions:

{q}_{1} \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; {q}_{2} \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \ell \in \{1, 2, \ldots, \max\!\left({q}_{1} {q}_{2}, 2\right) - 1\} \;\mathbin{\operatorname{and}}\; \gcd\!\left(\ell, {q}_{1}\right) = \gcd\!\left(\ell, {q}_{2}\right) = \gcd\!\left({q}_{1}, {q}_{2}\right) = 1

TeX:

\chi_{{q}_{1} {q}_{2} \, . \, \ell} = \chi_{{q}_{1} \, . \, \ell \bmod {q}_{1}} \chi_{{q}_{2} \, . \, \ell \bmod {q}_{2}}

{q}_{1} \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; {q}_{2} \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \ell \in \{1, 2, \ldots, \max\!\left({q}_{1} {q}_{2}, 2\right) - 1\} \;\mathbin{\operatorname{and}}\; \gcd\!\left(\ell, {q}_{1}\right) = \gcd\!\left(\ell, {q}_{2}\right) = \gcd\!\left({q}_{1}, {q}_{2}\right) = 1

Definitions:

Fungrim symbol	Notation	Short description
`DirichletCharacter`	$\chi_{q \, . \, \ell}$	Dirichlet character
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`Range`	$\{a, a + 1, \ldots, b\}$	Integers between given endpoints
`GCD`	$\gcd\!\left(a, b\right)$	Greatest common divisor

Source code for this entry:

Entry(ID("2a48bd"),
    Formula(Equal(DirichletCharacter(Mul(Subscript(q, 1), Subscript(q, 2)), ell), Mul(DirichletCharacter(Subscript(q, 1), Mod(ell, Subscript(q, 1))), DirichletCharacter(Subscript(q, 2), Mod(ell, Subscript(q, 2)))))),
    Variables(Subscript(q, 1), Subscript(q, 2), ell),
    Assumptions(And(Element(Subscript(q, 1), ZZGreaterEqual(1)), Element(Subscript(q, 2), ZZGreaterEqual(1)), Element(ell, Range(1, Sub(Max(Mul(Subscript(q, 1), Subscript(q, 2)), 2), 1))), Equal(GCD(ell, Subscript(q, 1)), GCD(ell, Subscript(q, 2)), GCD(Subscript(q, 1), Subscript(q, 2)), 1))))

166a87

Symbol: ConreyGenerator —

g_{p}

— Conrey generator

For an odd prime

p

, the Conrey generator is defined as the smallest positive integer

g_{p}

that generates the multiplicative group of order

{p}^{e}

for every

e \ge 1

.

Definitions:

Fungrim symbol	Notation	Short description
`ConreyGenerator`	$g_{p}$	Conrey generator
`Pow`	${a}^{b}$	Power

Source code for this entry:

Entry(ID("166a87"),
    SymbolDefinition(ConreyGenerator, ConreyGenerator(p), "Conrey generator"),
    Description("For an odd prime", p, ", the Conrey generator is defined as the smallest positive integer", ConreyGenerator(p), "that generates the multiplicative group of order", Pow(p, e), "for every", GreaterEqual(e, 1), "."))

a9847a

Symbol: DiscreteLog —

\log_{b}\!\left(x\right) \bmod q

— Discrete logarithm

\log_{b}\!\left(x\right) \bmod q

represents the smallest positive integer

y

such that

{b}^{y} \equiv x \pmod {q}

Definitions:

Fungrim symbol	Notation	Short description
`DiscreteLog`	$\log_{b}\!\left(x\right) \bmod q$	Discrete logarithm
`Pow`	${a}^{b}$	Power

Source code for this entry:

Entry(ID("a9847a"),
    SymbolDefinition(DiscreteLog, DiscreteLog(x, b, q), "Discrete logarithm"),
    Description(DiscreteLog(x, b, q), "represents the smallest positive integer", y, "such that", CongruentMod(Pow(b, y), x, q)))

75231e

g_{p} = \min \left\{ a : a \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \# \left\{ {a}^{k} \bmod p : k \in \mathbb{Z}_{\ge 0} \right\} = p - 1 \;\mathbin{\operatorname{and}}\; \# \left\{ {a}^{k} \bmod {p}^{2} : k \in \mathbb{Z}_{\ge 0} \right\} = p \left(p - 1\right) \right\}

Assumptions:

p \in \mathbb{P} \;\mathbin{\operatorname{and}}\; p \ge 3

TeX:

g_{p} = \min \left\{ a : a \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \# \left\{ {a}^{k} \bmod p : k \in \mathbb{Z}_{\ge 0} \right\} = p - 1 \;\mathbin{\operatorname{and}}\; \# \left\{ {a}^{k} \bmod {p}^{2} : k \in \mathbb{Z}_{\ge 0} \right\} = p \left(p - 1\right) \right\}

p \in \mathbb{P} \;\mathbin{\operatorname{and}}\; p \ge 3

Definitions:

Fungrim symbol	Notation	Short description
`ConreyGenerator`	$g_{p}$	Conrey generator
`Minimum`	$\mathop{\min}\limits_{x \in S} f(x)$	Minimum value of a set or function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`Cardinality`	$\# S$	Set cardinality
`Pow`	${a}^{b}$	Power
`PP`	$\mathbb{P}$	Prime numbers

Source code for this entry:

Entry(ID("75231e"),
    Formula(Equal(ConreyGenerator(p), Minimum(Set(a, For(a), And(Element(a, ZZGreaterEqual(1)), Equal(Cardinality(Set(Mod(Pow(a, k), p), For(k), Element(k, ZZGreaterEqual(0)))), Sub(p, 1)), Equal(Cardinality(Set(Mod(Pow(a, k), Pow(p, 2)), For(k), Element(k, ZZGreaterEqual(0)))), Mul(p, Sub(p, 1)))))))),
    Variables(p),
    Assumptions(And(Element(p, PP), GreaterEqual(p, 3))))

540931

g_{p} = \begin{cases} 10, & p = 40487\\7, & p = 6692367337\\\min\left(A\right), & \text{otherwise}\\ \end{cases}\; \text{ where } A = \left\{ a : a \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \# \left\{ {a}^{k} \bmod p : k \in \mathbb{Z}_{\ge 0} \right\} = p - 1 \right\}

Assumptions:

p \in \mathbb{P} \;\mathbin{\operatorname{and}}\; p \ge 3 \;\mathbin{\operatorname{and}}\; p < {10}^{12}

TeX:

g_{p} = \begin{cases} 10, & p = 40487\\7, & p = 6692367337\\\min\left(A\right), & \text{otherwise}\\ \end{cases}\; \text{ where } A = \left\{ a : a \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \# \left\{ {a}^{k} \bmod p : k \in \mathbb{Z}_{\ge 0} \right\} = p - 1 \right\}

p \in \mathbb{P} \;\mathbin{\operatorname{and}}\; p \ge 3 \;\mathbin{\operatorname{and}}\; p < {10}^{12}

Definitions:

Fungrim symbol	Notation	Short description
`ConreyGenerator`	$g_{p}$	Conrey generator
`Minimum`	$\mathop{\min}\limits_{x \in S} f(x)$	Minimum value of a set or function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`Cardinality`	$\# S$	Set cardinality
`Pow`	${a}^{b}$	Power
`PP`	$\mathbb{P}$	Prime numbers

Source code for this entry:

Entry(ID("540931"),
    Formula(Where(Equal(ConreyGenerator(p), Cases(Tuple(10, Equal(p, 40487)), Tuple(7, Equal(p, 6692367337)), Tuple(Minimum(A), Otherwise))), Equal(A, Set(a, For(a), And(Element(a, ZZGreaterEqual(1)), Equal(Cardinality(Set(Mod(Pow(a, k), p), For(k), Element(k, ZZGreaterEqual(0)))), Sub(p, 1))))))),
    Variables(p),
    Assumptions(And(Element(p, PP), GreaterEqual(p, 3), Less(p, Pow(10, 12)))))

4cf4e4

\chi_{q \, . \, \ell}\!\left(n\right) = \exp\!\left(\frac{2 \pi i a b}{\varphi(q)}\right)\; \text{ where } q = {p}^{e},\;g = g_{p},\;a = \log_{g}\!\left(\ell\right) \bmod q,\;b = \log_{g}\!\left(n\right) \bmod q

Assumptions:

p \in \mathbb{P} \;\mathbin{\operatorname{and}}\; p \ge 3 \;\mathbin{\operatorname{and}}\; e \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \ell \in \{1, 2, \ldots, {p}^{e} - 1\} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; \gcd\!\left(\ell, {p}^{e}\right) = \gcd\!\left(n, {p}^{e}\right) = 1

TeX:

\chi_{q \, . \, \ell}\!\left(n\right) = \exp\!\left(\frac{2 \pi i a b}{\varphi(q)}\right)\; \text{ where } q = {p}^{e},\;g = g_{p},\;a = \log_{g}\!\left(\ell\right) \bmod q,\;b = \log_{g}\!\left(n\right) \bmod q

p \in \mathbb{P} \;\mathbin{\operatorname{and}}\; p \ge 3 \;\mathbin{\operatorname{and}}\; e \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \ell \in \{1, 2, \ldots, {p}^{e} - 1\} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; \gcd\!\left(\ell, {p}^{e}\right) = \gcd\!\left(n, {p}^{e}\right) = 1

Definitions:

Fungrim symbol	Notation	Short description
`DirichletCharacter`	$\chi_{q \, . \, \ell}$	Dirichlet character
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`Totient`	$\varphi(n)$	Euler totient function
`Pow`	${a}^{b}$	Power
`ConreyGenerator`	$g_{p}$	Conrey generator
`DiscreteLog`	$\log_{b}\!\left(x\right) \bmod q$	Discrete logarithm
`PP`	$\mathbb{P}$	Prime numbers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`Range`	$\{a, a + 1, \ldots, b\}$	Integers between given endpoints
`ZZ`	$\mathbb{Z}$	Integers
`GCD`	$\gcd\!\left(a, b\right)$	Greatest common divisor

Source code for this entry:

Entry(ID("4cf4e4"),
    Formula(Where(Equal(DirichletCharacter(q, ell, n), Exp(Div(Mul(Mul(Mul(Mul(2, Pi), ConstI), a), b), Totient(q)))), Equal(q, Pow(p, e)), Equal(g, ConreyGenerator(p)), Equal(a, DiscreteLog(ell, g, q)), Equal(b, DiscreteLog(n, g, q)))),
    Variables(p, e, ell, n),
    Assumptions(And(Element(p, PP), GreaterEqual(p, 3), Element(e, ZZGreaterEqual(1)), Element(ell, Range(1, Sub(Pow(p, e), 1))), Element(n, ZZ), Equal(GCD(ell, Pow(p, e)), GCD(n, Pow(p, e)), 1))))

fc4f6a

\chi_{4 \, . \, 3}\!\left(n\right) = \begin{cases} 1, & n \equiv 1 \pmod {4}\\-1, & n \equiv 3 \pmod {4}\\0, & \text{otherwise}\\ \end{cases}

Assumptions:

n \in \mathbb{Z}

TeX:

\chi_{4 \, . \, 3}\!\left(n\right) = \begin{cases} 1, & n \equiv 1 \pmod {4}\\-1, & n \equiv 3 \pmod {4}\\0, & \text{otherwise}\\ \end{cases}

n \in \mathbb{Z}

Definitions:

Fungrim symbol	Notation	Short description
`DirichletCharacter`	$\chi_{q \, . \, \ell}$	Dirichlet character
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("fc4f6a"),
    Formula(Equal(DirichletCharacter(4, 3, n), Cases(Tuple(1, CongruentMod(n, 1, 4)), Tuple(-1, CongruentMod(n, 3, 4)), Tuple(0, Otherwise)))),
    Variables(n),
    Assumptions(Element(n, ZZ)))

03fbe8

\chi_{q \, . \, \ell}\!\left(n\right) = \exp\!\left(2 \pi i \left(\frac{\left(1 - x\right) \left(1 - y\right)}{8} + \frac{a b}{{2}^{e - 2}}\right)\right)\; \text{ where } q = {2}^{e},\;L(k) = \begin{cases} \left(1, \log_{5}\!\left(k\right) \bmod q\right), & k \in \left\{ {5}^{i} \bmod q : i \in \mathbb{Z}_{\ge 1} \right\}\\\left(-1, \log_{5}\!\left(-k\right) \bmod q\right), & k \in \left\{ -{5}^{i} \bmod q : i \in \mathbb{Z}_{\ge 1} \right\}\\ \end{cases},\;\left(x, a\right) = L(\ell),\;\left(y, b\right) = L(n)

Assumptions:

e \in \mathbb{Z}_{\ge 3} \;\mathbin{\operatorname{and}}\; \ell \in \{1, 2, \ldots, {2}^{e} - 1\} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; \gcd\!\left(\ell, 2\right) = \gcd\!\left(n, 2\right) = 1

TeX:

\chi_{q \, . \, \ell}\!\left(n\right) = \exp\!\left(2 \pi i \left(\frac{\left(1 - x\right) \left(1 - y\right)}{8} + \frac{a b}{{2}^{e - 2}}\right)\right)\; \text{ where } q = {2}^{e},\;L(k) = \begin{cases} \left(1, \log_{5}\!\left(k\right) \bmod q\right), & k \in \left\{ {5}^{i} \bmod q : i \in \mathbb{Z}_{\ge 1} \right\}\\\left(-1, \log_{5}\!\left(-k\right) \bmod q\right), & k \in \left\{ -{5}^{i} \bmod q : i \in \mathbb{Z}_{\ge 1} \right\}\\ \end{cases},\;\left(x, a\right) = L(\ell),\;\left(y, b\right) = L(n)

e \in \mathbb{Z}_{\ge 3} \;\mathbin{\operatorname{and}}\; \ell \in \{1, 2, \ldots, {2}^{e} - 1\} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; \gcd\!\left(\ell, 2\right) = \gcd\!\left(n, 2\right) = 1

Definitions:

Fungrim symbol	Notation	Short description
`DirichletCharacter`	$\chi_{q \, . \, \ell}$	Dirichlet character
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`Pow`	${a}^{b}$	Power
`DiscreteLog`	$\log_{b}\!\left(x\right) \bmod q$	Discrete logarithm
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`Range`	$\{a, a + 1, \ldots, b\}$	Integers between given endpoints
`ZZ`	$\mathbb{Z}$	Integers
`GCD`	$\gcd\!\left(a, b\right)$	Greatest common divisor

Source code for this entry:

Entry(ID("03fbe8"),
    Formula(Where(Equal(DirichletCharacter(q, ell, n), Exp(Mul(Mul(Mul(2, Pi), ConstI), Add(Div(Mul(Sub(1, x), Sub(1, y)), 8), Div(Mul(a, b), Pow(2, Sub(e, 2))))))), Equal(q, Pow(2, e)), Equal(L(k), Cases(Tuple(Tuple(1, DiscreteLog(k, 5, q)), Element(k, Set(Mod(Pow(5, i), q), For(i), Element(i, ZZGreaterEqual(1))))), Tuple(Tuple(-1, DiscreteLog(Neg(k), 5, q)), Element(k, Set(Mod(Neg(Pow(5, i)), q), For(i), Element(i, ZZGreaterEqual(1))))))), Equal(Tuple(x, a), L(ell)), Equal(Tuple(y, b), L(n)))),
    Variables(e, ell, n),
    Assumptions(And(Element(e, ZZGreaterEqual(3)), Element(ell, Range(1, Sub(Pow(2, e), 1))), Element(n, ZZ), Equal(GCD(ell, 2), GCD(n, 2), 1))))

4877d1

\sum_{n=0}^{q - 1} \chi(n) = \begin{cases} \varphi(q), & \chi = \chi_{q \, . \, 1}\\0, & \text{otherwise}\\ \end{cases}

Assumptions:

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G_{q}

TeX:

\sum_{n=0}^{q - 1} \chi(n) = \begin{cases} \varphi(q), & \chi = \chi_{q \, . \, 1}\\0, & \text{otherwise}\\ \end{cases}

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G_{q}

Definitions:

Fungrim symbol	Notation	Short description
`Sum`	$\sum_{n} f(n)$	Sum
`Totient`	$\varphi(n)$	Euler totient function
`DirichletCharacter`	$\chi_{q \, . \, \ell}$	Dirichlet character
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`DirichletGroup`	$G_{q}$	Dirichlet characters with given modulus

Source code for this entry:

Entry(ID("4877d1"),
    Formula(Equal(Sum(chi(n), For(n, 0, Sub(q, 1))), Cases(Tuple(Totient(q), Equal(chi, DirichletCharacter(q, 1))), Tuple(0, Otherwise)))),
    Variables(q, chi),
    Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(chi, DirichletGroup(q)))))

3ab92d

\sum_{\chi \in G_{q}} \chi(n) = \begin{cases} \varphi(q), & n \equiv 1 \pmod {q}\\0, & \text{otherwise}\\ \end{cases}

Assumptions:

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}

TeX:

\sum_{\chi \in G_{q}} \chi(n) = \begin{cases} \varphi(q), & n \equiv 1 \pmod {q}\\0, & \text{otherwise}\\ \end{cases}

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}

Definitions:

Fungrim symbol	Notation	Short description
`Sum`	$\sum_{n} f(n)$	Sum
`DirichletGroup`	$G_{q}$	Dirichlet characters with given modulus
`Totient`	$\varphi(n)$	Euler totient function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("3ab92d"),
    Formula(Equal(Sum(chi(n), ForElement(chi, DirichletGroup(q))), Cases(Tuple(Totient(q), CongruentMod(n, 1, q)), Tuple(0, Otherwise)))),
    Variables(q, n),
    Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(n, ZZ))))

a4e947

\sum_{n=0}^{q - 1} {\chi}_{1}(n) \overline{{\chi}_{2}(n)} = \begin{cases} \varphi(q), & {\chi}_{1} = {\chi}_{2}\\0, & \text{otherwise}\\ \end{cases}

Assumptions:

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; {\chi}_{1} \in G_{q} \;\mathbin{\operatorname{and}}\; {\chi}_{2} \in G_{q}

TeX:

\sum_{n=0}^{q - 1} {\chi}_{1}(n) \overline{{\chi}_{2}(n)} = \begin{cases} \varphi(q), & {\chi}_{1} = {\chi}_{2}\\0, & \text{otherwise}\\ \end{cases}

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; {\chi}_{1} \in G_{q} \;\mathbin{\operatorname{and}}\; {\chi}_{2} \in G_{q}

Definitions:

Fungrim symbol	Notation	Short description
`Sum`	$\sum_{n} f(n)$	Sum
`Conjugate`	$\overline{z}$	Complex conjugate
`Totient`	$\varphi(n)$	Euler totient function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`DirichletGroup`	$G_{q}$	Dirichlet characters with given modulus

Source code for this entry:

Entry(ID("a4e947"),
    Formula(Equal(Sum(Mul(Subscript(chi, 1)(n), Conjugate(Subscript(chi, 2)(n))), For(n, 0, Sub(q, 1))), Cases(Tuple(Totient(q), Equal(Subscript(chi, 1), Subscript(chi, 2))), Tuple(0, Otherwise)))),
    Variables(q, Subscript(chi, 1), Subscript(chi, 2)),
    Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(Subscript(chi, 1), DirichletGroup(q)), Element(Subscript(chi, 2), DirichletGroup(q)))))

f4de66

\sum_{\chi \in G_{q}} \chi(m) \overline{\chi(n)} = \begin{cases} \varphi(q), & n \equiv m \pmod {q} \;\mathbin{\operatorname{and}}\; \gcd\!\left(m, q\right) = 1\\0, & \text{otherwise}\\ \end{cases}

Assumptions:

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; m \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}

TeX:

\sum_{\chi \in G_{q}} \chi(m) \overline{\chi(n)} = \begin{cases} \varphi(q), & n \equiv m \pmod {q} \;\mathbin{\operatorname{and}}\; \gcd\!\left(m, q\right) = 1\\0, & \text{otherwise}\\ \end{cases}

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; m \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}

Definitions:

Fungrim symbol	Notation	Short description
`Sum`	$\sum_{n} f(n)$	Sum
`Conjugate`	$\overline{z}$	Complex conjugate
`DirichletGroup`	$G_{q}$	Dirichlet characters with given modulus
`Totient`	$\varphi(n)$	Euler totient function
`GCD`	$\gcd\!\left(a, b\right)$	Greatest common divisor
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("f4de66"),
    Formula(Equal(Sum(Mul(chi(m), Conjugate(chi(n))), ForElement(chi, DirichletGroup(q))), Cases(Tuple(Totient(q), And(CongruentMod(n, m, q), Equal(GCD(m, q), 1))), Tuple(0, Otherwise)))),
    Variables(q, m, n),
    Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(m, ZZ), Element(n, ZZ))))

338b5c

Table of

G^{\text{Primitive}}_{q}

and

G_{q} \setminus G^{\text{Primitive}}_{q}

for

1 \le q \le 30

$q$	$\# G_{q}$	$\# G^{\text{Primitive}}_{q}$	$\ell$ such that $\chi_{q \, . \, \ell}$ is primitive	$\ell$ such that $\chi_{q \, . \, \ell}$ is non-primitive
1	1	1	{1}	{}
2	1	0	{}	{1}
3	2	1	{2}	{1}
4	2	1	{3}	{1}
5	4	3	{2, 3, 4}	{1}
6	2	0	{}	{1, 5}
7	6	5	{2, 3, 4, 5, 6}	{1}
8	4	2	{3, 5}	{1, 7}
9	6	4	{2, 4, 5, 7}	{1, 8}
10	4	0	{}	{1, 3, 7, 9}
11	10	9	{2, 3, 4, 5, 6, 7, 8, 9, 10}	{1}
12	4	1	{11}	{1, 5, 7}
13	12	11	{2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}	{1}
14	6	0	{}	{1, 3, 5, 9, 11, 13}
15	8	3	{2, 8, 14}	{1, 4, 7, 11, 13}
16	8	4	{3, 5, 11, 13}	{1, 7, 9, 15}
17	16	15	{2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}	{1}
18	6	0	{}	{1, 5, 7, 11, 13, 17}
19	18	17	{2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18}	{1}
20	8	3	{3, 7, 19}	{1, 9, 11, 13, 17}
21	12	5	{2, 5, 11, 17, 20}	{1, 4, 8, 10, 13, 16, 19}
22	10	0	{}	{1, 3, 5, 7, 9, 13, 15, 17, 19, 21}
23	22	21	{2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22}	{1}
24	8	2	{5, 11}	{1, 7, 13, 17, 19, 23}
25	20	16	{2, 3, 4, 6, 8, 9, 11, 12, 13, 14, 16, 17, 19, 21, 22, 23}	{1, 7, 18, 24}
26	12	0	{}	{1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25}
27	18	12	{2, 4, 5, 7, 11, 13, 14, 16, 20, 22, 23, 25}	{1, 8, 10, 17, 19, 26}
28	12	5	{3, 11, 19, 23, 27}	{1, 5, 9, 13, 15, 17, 25}
29	28	27	{2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28}	{1}
30	8	0	{}	{1, 7, 11, 13, 17, 19, 23, 29}

Table data:

\left(q, \# G_{q}, \# G^{\text{Primitive}}_{q}, P, N\right)

such that

G^{\text{Primitive}}_{q} = \left\{ \chi_{q \, . \, \ell} : \ell \in P \right\} \;\mathbin{\operatorname{and}}\; G_{q} \setminus G^{\text{Primitive}}_{q} = \left\{ \chi_{q \, . \, \ell} : \ell \in N \right\}

Definitions:

Fungrim symbol	Notation	Short description
`PrimitiveDirichletCharacters`	$G^{\text{Primitive}}_{q}$	Primitive Dirichlet characters with given modulus
`DirichletGroup`	$G_{q}$	Dirichlet characters with given modulus
`Cardinality`	$\# S$	Set cardinality
`DirichletCharacter`	$\chi_{q \, . \, \ell}$	Dirichlet character

Source code for this entry:

Entry(ID("338b5c"),
    Description("Table of", PrimitiveDirichletCharacters(q), "and", SetMinus(DirichletGroup(q), PrimitiveDirichletCharacters(q)), "for", LessEqual(1, q, 30)),
    Table(TableRelation(Tuple(q, Cardinality(DirichletGroup(q)), Cardinality(PrimitiveDirichletCharacters(q)), P, N), And(Equal(PrimitiveDirichletCharacters(q), Set(DirichletCharacter(q, ell), For(ell), Element(ell, P))), Equal(SetMinus(DirichletGroup(q), PrimitiveDirichletCharacters(q)), Set(DirichletCharacter(q, ell), For(ell), Element(ell, N))))), TableHeadings(q, Cardinality(DirichletGroup(q)), Cardinality(PrimitiveDirichletCharacters(q)), Description(ell, "such that", DirichletCharacter(q, ell), "is primitive"), Description(ell, "such that", DirichletCharacter(q, ell), "is non-primitive")), List(Tuple(1, 1, 1, Set(1), Set()), Tuple(2, 1, 0, Set(), Set(1)), Tuple(3, 2, 1, Set(2), Set(1)), Tuple(4, 2, 1, Set(3), Set(1)), Tuple(5, 4, 3, Set(2, 3, 4), Set(1)), Tuple(6, 2, 0, Set(), Set(1, 5)), Tuple(7, 6, 5, Set(2, 3, 4, 5, 6), Set(1)), Tuple(8, 4, 2, Set(3, 5), Set(1, 7)), Tuple(9, 6, 4, Set(2, 4, 5, 7), Set(1, 8)), Tuple(10, 4, 0, Set(), Set(1, 3, 7, 9)), Tuple(11, 10, 9, Set(2, 3, 4, 5, 6, 7, 8, 9, 10), Set(1)), Tuple(12, 4, 1, Set(11), Set(1, 5, 7)), Tuple(13, 12, 11, Set(2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12), Set(1)), Tuple(14, 6, 0, Set(), Set(1, 3, 5, 9, 11, 13)), Tuple(15, 8, 3, Set(2, 8, 14), Set(1, 4, 7, 11, 13)), Tuple(16, 8, 4, Set(3, 5, 11, 13), Set(1, 7, 9, 15)), Tuple(17, 16, 15, Set(2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16), Set(1)), Tuple(18, 6, 0, Set(), Set(1, 5, 7, 11, 13, 17)), Tuple(19, 18, 17, Set(2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18), Set(1)), Tuple(20, 8, 3, Set(3, 7, 19), Set(1, 9, 11, 13, 17)), Tuple(21, 12, 5, Set(2, 5, 11, 17, 20), Set(1, 4, 8, 10, 13, 16, 19)), Tuple(22, 10, 0, Set(), Set(1, 3, 5, 7, 9, 13, 15, 17, 19, 21)), Tuple(23, 22, 21, Set(2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22), Set(1)), Tuple(24, 8, 2, Set(5, 11), Set(1, 7, 13, 17, 19, 23)), Tuple(25, 20, 16, Set(2, 3, 4, 6, 8, 9, 11, 12, 13, 14, 16, 17, 19, 21, 22, 23), Set(1, 7, 18, 24)), Tuple(26, 12, 0, Set(), Set(1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25)), Tuple(27, 18, 12, Set(2, 4, 5, 7, 11, 13, 14, 16, 20, 22, 23, 25), Set(1, 8, 10, 17, 19, 26)), Tuple(28, 12, 5, Set(3, 11, 19, 23, 27), Set(1, 5, 9, 13, 15, 17, 25)), Tuple(29, 28, 27, Set(2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28), Set(1)), Tuple(30, 8, 0, Set(), Set(1, 7, 11, 13, 17, 19, 23, 29)))))

c40df4

Table of

\chi_{1 \, . \, \ell}

$\ell$ \ $n$	0
1	1

Table data:

\left(\ell, n, y\right)

such that

\chi(n) = y\; \text{ where } \chi = \chi_{1 \, . \, \ell}

Definitions:

Fungrim symbol	Notation	Short description
`DirichletCharacter`	$\chi_{q \, . \, \ell}$	Dirichlet character

Source code for this entry:

Entry(ID("c40df4"),
    Description("Table of", DirichletCharacter(1, ell)),
    Table(TableRelation(Tuple(ell, n, y), Where(Equal(chi(n), y), Equal(chi, DirichletCharacter(1, ell)))), TableHeadings(Description(ell, "\", n), 0), TableColumnHeadings(1), List(Tuple(1))))

ef432d

Table of

\chi_{2 \, . \, \ell}

$\ell$ \ $n$	0	1
1	0	1

Table data:

\left(\ell, n, y\right)

such that

\chi(n) = y\; \text{ where } \chi = \chi_{2 \, . \, \ell}

Definitions:

Fungrim symbol	Notation	Short description
`DirichletCharacter`	$\chi_{q \, . \, \ell}$	Dirichlet character

Source code for this entry:

Entry(ID("ef432d"),
    Description("Table of", DirichletCharacter(2, ell)),
    Table(TableRelation(Tuple(ell, n, y), Where(Equal(chi(n), y), Equal(chi, DirichletCharacter(2, ell)))), TableHeadings(Description(ell, "\", n), 0, 1), TableColumnHeadings(1), List(Tuple(0, 1))))

287d9b

Table of

\chi_{3 \, . \, \ell}

$\ell$ \ $n$	0	1	2
1	0	1	1
2	0	1	-1

Table data:

\left(\ell, n, y\right)

such that

\chi(n) = y\; \text{ where } \chi = \chi_{3 \, . \, \ell}

Definitions:

Fungrim symbol	Notation	Short description
`DirichletCharacter`	$\chi_{q \, . \, \ell}$	Dirichlet character

Source code for this entry:

Entry(ID("287d9b"),
    Description("Table of", DirichletCharacter(3, ell)),
    Table(TableRelation(Tuple(ell, n, y), Where(Equal(chi(n), y), Equal(chi, DirichletCharacter(3, ell)))), TableHeadings(Description(ell, "\", n), 0, 1, 2), TableColumnHeadings(1, 2), List(Tuple(0, 1, 1), Tuple(0, 1, -1))))

5dc1c0

Table of

\chi_{4 \, . \, \ell}

$\ell$ \ $n$	0	1	2	3
1	0	1	0	1
3	0	1	0	-1

Table data:

\left(\ell, n, y\right)

such that

\chi(n) = y\; \text{ where } \chi = \chi_{4 \, . \, \ell}

Definitions:

Fungrim symbol	Notation	Short description
`DirichletCharacter`	$\chi_{q \, . \, \ell}$	Dirichlet character

Source code for this entry:

Entry(ID("5dc1c0"),
    Description("Table of", DirichletCharacter(4, ell)),
    Table(TableRelation(Tuple(ell, n, y), Where(Equal(chi(n), y), Equal(chi, DirichletCharacter(4, ell)))), TableHeadings(Description(ell, "\", n), 0, 1, 2, 3), TableColumnHeadings(1, 3), List(Tuple(0, 1, 0, 1), Tuple(0, 1, 0, -1))))

668877

Table of

\chi_{5 \, . \, \ell}

$\ell$ \ $n$	1	2	3	4
1	1	1	1	1
2	1	$i$	$-i$	-1
3	1	$-i$	$i$	-1
4	1	-1	-1	1

Table data:

\left(\ell, n, y\right)

such that

\chi(n) = y\; \text{ where } \chi = \chi_{5 \, . \, \ell}

Definitions:

Fungrim symbol	Notation	Short description
`DirichletCharacter`	$\chi_{q \, . \, \ell}$	Dirichlet character
`ConstI`	$i$	Imaginary unit

Source code for this entry:

Entry(ID("668877"),
    Description("Table of", DirichletCharacter(5, ell)),
    Table(TableRelation(Tuple(ell, n, y), Where(Equal(chi(n), y), Equal(chi, DirichletCharacter(5, ell)))), TableHeadings(Description(ell, "\", n), 0, 1, 2, 3, 4), TableColumnHeadings(1, 2, 3, 4), List(Tuple(0, 1, 1, 1, 1), Tuple(0, 1, ConstI, Neg(ConstI), -1), Tuple(0, 1, Neg(ConstI), ConstI, -1), Tuple(0, 1, -1, -1, 1))))

d8cac6

Table of

\chi_{6 \, . \, \ell}

$\ell$ \ $n$	0	1	2	3	4	5
1	0	1	0	0	0	1
5	0	1	0	0	0	-1

Table data:

\left(\ell, n, y\right)

such that

\chi(n) = y\; \text{ where } \chi = \chi_{6 \, . \, \ell}

Definitions:

Fungrim symbol	Notation	Short description
`DirichletCharacter`	$\chi_{q \, . \, \ell}$	Dirichlet character

Source code for this entry:

Entry(ID("d8cac6"),
    Description("Table of", DirichletCharacter(6, ell)),
    Table(TableRelation(Tuple(ell, n, y), Where(Equal(chi(n), y), Equal(chi, DirichletCharacter(6, ell)))), TableHeadings(Description(ell, "\", n), 0, 1, 2, 3, 4, 5), TableColumnHeadings(1, 5), List(Tuple(0, 1, 0, 0, 0, 1), Tuple(0, 1, 0, 0, 0, -1))))

ec0054

Table of

\chi_{7 \, . \, \ell}

$\ell$ \ $n$	1	2	3	4	5	6
1	1	1	1	1	1	1
2	1	$-{e}^{\pi i / 3}$	${e}^{2 \pi i / 3}$	${e}^{2 \pi i / 3}$	$-{e}^{\pi i / 3}$	1
3	1	${e}^{2 \pi i / 3}$	${e}^{\pi i / 3}$	$-{e}^{\pi i / 3}$	$-{e}^{2 \pi i / 3}$	-1
4	1	${e}^{2 \pi i / 3}$	$-{e}^{\pi i / 3}$	$-{e}^{\pi i / 3}$	${e}^{2 \pi i / 3}$	1
5	1	$-{e}^{\pi i / 3}$	$-{e}^{2 \pi i / 3}$	${e}^{2 \pi i / 3}$	${e}^{\pi i / 3}$	-1
6	1	1	-1	1	-1	-1

Table data:

\left(\ell, n, y\right)

such that

\chi(n) = y\; \text{ where } \chi = \chi_{7 \, . \, \ell}

Definitions:

Fungrim symbol	Notation	Short description
`DirichletCharacter`	$\chi_{q \, . \, \ell}$	Dirichlet character
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit

Source code for this entry:

Entry(ID("ec0054"),
    Description("Table of", DirichletCharacter(7, ell)),
    Table(TableRelation(Tuple(ell, n, y), Where(Equal(chi(n), y), Equal(chi, DirichletCharacter(7, ell)))), TableHeadings(Description(ell, "\", n), 0, 1, 2, 3, 4, 5, 6), TableColumnHeadings(1, 2, 3, 4, 5, 6), List(Tuple(0, 1, 1, 1, 1, 1, 1), Tuple(0, 1, Neg(Exp(Div(Mul(Pi, ConstI), 3))), Exp(Div(Mul(Mul(2, Pi), ConstI), 3)), Exp(Div(Mul(Mul(2, Pi), ConstI), 3)), Neg(Exp(Div(Mul(Pi, ConstI), 3))), 1), Tuple(0, 1, Exp(Div(Mul(Mul(2, Pi), ConstI), 3)), Exp(Div(Mul(Pi, ConstI), 3)), Neg(Exp(Div(Mul(Pi, ConstI), 3))), Neg(Exp(Div(Mul(Mul(2, Pi), ConstI), 3))), -1), Tuple(0, 1, Exp(Div(Mul(Mul(2, Pi), ConstI), 3)), Neg(Exp(Div(Mul(Pi, ConstI), 3))), Neg(Exp(Div(Mul(Pi, ConstI), 3))), Exp(Div(Mul(Mul(2, Pi), ConstI), 3)), 1), Tuple(0, 1, Neg(Exp(Div(Mul(Pi, ConstI), 3))), Neg(Exp(Div(Mul(Mul(2, Pi), ConstI), 3))), Exp(Div(Mul(Mul(2, Pi), ConstI), 3)), Exp(Div(Mul(Pi, ConstI), 3)), -1), Tuple(0, 1, 1, -1, 1, -1, -1))))

fc267b

Table of

\chi_{8 \, . \, \ell}

$\ell$ \ $n$	1	3	5	7
1	1	1	1	1
3	1	1	-1	-1
5	1	-1	-1	1
7	1	-1	1	-1

Table data:

\left(\ell, n, y\right)

such that

\chi(n) = y\; \text{ where } \chi = \chi_{8 \, . \, \ell}

Definitions:

Fungrim symbol	Notation	Short description
`DirichletCharacter`	$\chi_{q \, . \, \ell}$	Dirichlet character

Source code for this entry:

Entry(ID("fc267b"),
    Description("Table of", DirichletCharacter(8, ell)),
    Table(TableRelation(Tuple(ell, n, y), Where(Equal(chi(n), y), Equal(chi, DirichletCharacter(8, ell)))), TableHeadings(Description(ell, "\", n), 0, 1, 2, 3, 4, 5, 6, 7), TableColumnHeadings(1, 3, 5, 7), List(Tuple(0, 1, 0, 1, 0, 1, 0, 1), Tuple(0, 1, 0, 1, 0, -1, 0, -1), Tuple(0, 1, 0, -1, 0, -1, 0, 1), Tuple(0, 1, 0, -1, 0, 1, 0, -1))))

0c7de4

Table of

\chi_{9 \, . \, \ell}

$\ell$ \ $n$	1	2	4	5	7	8
1	1	1	1	1	1	1
2	1	${e}^{\pi i / 3}$	${e}^{2 \pi i / 3}$	$-{e}^{2 \pi i / 3}$	$-{e}^{\pi i / 3}$	-1
4	1	${e}^{2 \pi i / 3}$	$-{e}^{\pi i / 3}$	$-{e}^{\pi i / 3}$	${e}^{2 \pi i / 3}$	1
5	1	$-{e}^{2 \pi i / 3}$	$-{e}^{\pi i / 3}$	${e}^{\pi i / 3}$	${e}^{2 \pi i / 3}$	-1
7	1	$-{e}^{\pi i / 3}$	${e}^{2 \pi i / 3}$	${e}^{2 \pi i / 3}$	$-{e}^{\pi i / 3}$	1
8	1	-1	1	-1	1	-1

Table data:

\left(\ell, n, y\right)

such that

\chi(n) = y\; \text{ where } \chi = \chi_{9 \, . \, \ell}

Definitions:

Fungrim symbol	Notation	Short description
`DirichletCharacter`	$\chi_{q \, . \, \ell}$	Dirichlet character
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit

Source code for this entry:

Entry(ID("0c7de4"),
    Description("Table of", DirichletCharacter(9, ell)),
    Table(TableRelation(Tuple(ell, n, y), Where(Equal(chi(n), y), Equal(chi, DirichletCharacter(9, ell)))), TableHeadings(Description(ell, "\", n), 0, 1, 2, 3, 4, 5, 6, 7, 8), TableColumnHeadings(1, 2, 4, 5, 7, 8), List(Tuple(0, 1, 1, 0, 1, 1, 0, 1, 1), Tuple(0, 1, Exp(Div(Mul(Pi, ConstI), 3)), 0, Exp(Div(Mul(Mul(2, Pi), ConstI), 3)), Neg(Exp(Div(Mul(Mul(2, Pi), ConstI), 3))), 0, Neg(Exp(Div(Mul(Pi, ConstI), 3))), -1), Tuple(0, 1, Exp(Div(Mul(Mul(2, Pi), ConstI), 3)), 0, Neg(Exp(Div(Mul(Pi, ConstI), 3))), Neg(Exp(Div(Mul(Pi, ConstI), 3))), 0, Exp(Div(Mul(Mul(2, Pi), ConstI), 3)), 1), Tuple(0, 1, Neg(Exp(Div(Mul(Mul(2, Pi), ConstI), 3))), 0, Neg(Exp(Div(Mul(Pi, ConstI), 3))), Exp(Div(Mul(Pi, ConstI), 3)), 0, Exp(Div(Mul(Mul(2, Pi), ConstI), 3)), -1), Tuple(0, 1, Neg(Exp(Div(Mul(Pi, ConstI), 3))), 0, Exp(Div(Mul(Mul(2, Pi), ConstI), 3)), Exp(Div(Mul(Mul(2, Pi), ConstI), 3)), 0, Neg(Exp(Div(Mul(Pi, ConstI), 3))), 1), Tuple(0, 1, -1, 0, 1, -1, 0, 1, -1))))

d8155f

Table of

\chi_{10 \, . \, \ell}

$\ell$ \ $n$	1	3	7	9
1	1	1	1	1
3	1	$i$	$-i$	-1
7	1	$-i$	$i$	-1
9	1	-1	-1	1

Table data:

\left(\ell, n, y\right)

such that

\chi(n) = y\; \text{ where } \chi = \chi_{10 \, . \, \ell}

Definitions:

Fungrim symbol	Notation	Short description
`DirichletCharacter`	$\chi_{q \, . \, \ell}$	Dirichlet character
`ConstI`	$i$	Imaginary unit

Source code for this entry:

Entry(ID("d8155f"),
    Description("Table of", DirichletCharacter(10, ell)),
    Table(TableRelation(Tuple(ell, n, y), Where(Equal(chi(n), y), Equal(chi, DirichletCharacter(10, ell)))), TableHeadings(Description(ell, "\", n), 0, 1, 2, 3, 4, 5, 6, 7, 8, 9), TableColumnHeadings(1, 3, 7, 9), List(Tuple(0, 1, 0, 1, 0, 0, 0, 1, 0, 1), Tuple(0, 1, 0, ConstI, 0, 0, 0, Neg(ConstI), 0, -1), Tuple(0, 1, 0, Neg(ConstI), 0, 0, 0, ConstI, 0, -1), Tuple(0, 1, 0, -1, 0, 0, 0, -1, 0, 1))))

7a56c2

Table of

\chi_{11 \, . \, \ell}

$\ell$ \ $n$	1	2	3	4	5	6	7	8	9	10
1	1	1	1	1	1	1	1	1	1	1
2	1	${e}^{\pi i / 5}$	$-{e}^{3 \pi i / 5}$	${e}^{2 \pi i / 5}$	${e}^{4 \pi i / 5}$	$-{e}^{4 \pi i / 5}$	$-{e}^{2 \pi i / 5}$	${e}^{3 \pi i / 5}$	$-{e}^{\pi i / 5}$	-1
3	1	$-{e}^{3 \pi i / 5}$	${e}^{4 \pi i / 5}$	$-{e}^{\pi i / 5}$	${e}^{2 \pi i / 5}$	${e}^{2 \pi i / 5}$	$-{e}^{\pi i / 5}$	${e}^{4 \pi i / 5}$	$-{e}^{3 \pi i / 5}$	1
4	1	${e}^{2 \pi i / 5}$	$-{e}^{\pi i / 5}$	${e}^{4 \pi i / 5}$	$-{e}^{3 \pi i / 5}$	$-{e}^{3 \pi i / 5}$	${e}^{4 \pi i / 5}$	$-{e}^{\pi i / 5}$	${e}^{2 \pi i / 5}$	1
5	1	${e}^{4 \pi i / 5}$	${e}^{2 \pi i / 5}$	$-{e}^{3 \pi i / 5}$	$-{e}^{\pi i / 5}$	$-{e}^{\pi i / 5}$	$-{e}^{3 \pi i / 5}$	${e}^{2 \pi i / 5}$	${e}^{4 \pi i / 5}$	1
6	1	$-{e}^{4 \pi i / 5}$	${e}^{2 \pi i / 5}$	$-{e}^{3 \pi i / 5}$	$-{e}^{\pi i / 5}$	${e}^{\pi i / 5}$	${e}^{3 \pi i / 5}$	$-{e}^{2 \pi i / 5}$	${e}^{4 \pi i / 5}$	-1
7	1	$-{e}^{2 \pi i / 5}$	$-{e}^{\pi i / 5}$	${e}^{4 \pi i / 5}$	$-{e}^{3 \pi i / 5}$	${e}^{3 \pi i / 5}$	$-{e}^{4 \pi i / 5}$	${e}^{\pi i / 5}$	${e}^{2 \pi i / 5}$	-1
8	1	${e}^{3 \pi i / 5}$	${e}^{4 \pi i / 5}$	$-{e}^{\pi i / 5}$	${e}^{2 \pi i / 5}$	$-{e}^{2 \pi i / 5}$	${e}^{\pi i / 5}$	$-{e}^{4 \pi i / 5}$	$-{e}^{3 \pi i / 5}$	-1
9	1	$-{e}^{\pi i / 5}$	$-{e}^{3 \pi i / 5}$	${e}^{2 \pi i / 5}$	${e}^{4 \pi i / 5}$	${e}^{4 \pi i / 5}$	${e}^{2 \pi i / 5}$	$-{e}^{3 \pi i / 5}$	$-{e}^{\pi i / 5}$	1
10	1	-1	1	1	1	-1	-1	-1	1	-1

Table data:

\left(\ell, n, y\right)

such that

\chi(n) = y\; \text{ where } \chi = \chi_{11 \, . \, \ell}

Definitions:

Fungrim symbol	Notation	Short description
`DirichletCharacter`	$\chi_{q \, . \, \ell}$	Dirichlet character
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit

Source code for this entry:

Entry(ID("7a56c2"),
    Description("Table of", DirichletCharacter(11, ell)),
    Table(TableRelation(Tuple(ell, n, y), Where(Equal(chi(n), y), Equal(chi, DirichletCharacter(11, ell)))), TableHeadings(Description(ell, "\", n), 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10), TableColumnHeadings(1, 2, 3, 4, 5, 6, 7, 8, 9, 10), List(Tuple(0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1), Tuple(0, 1, Exp(Div(Mul(Pi, ConstI), 5)), Neg(Exp(Div(Mul(Mul(3, Pi), ConstI), 5))), Exp(Div(Mul(Mul(2, Pi), ConstI), 5)), Exp(Div(Mul(Mul(4, Pi), ConstI), 5)), Neg(Exp(Div(Mul(Mul(4, Pi), ConstI), 5))), Neg(Exp(Div(Mul(Mul(2, Pi), ConstI), 5))), Exp(Div(Mul(Mul(3, Pi), ConstI), 5)), Neg(Exp(Div(Mul(Pi, ConstI), 5))), -1), Tuple(0, 1, Neg(Exp(Div(Mul(Mul(3, Pi), ConstI), 5))), Exp(Div(Mul(Mul(4, Pi), ConstI), 5)), Neg(Exp(Div(Mul(Pi, ConstI), 5))), Exp(Div(Mul(Mul(2, Pi), ConstI), 5)), Exp(Div(Mul(Mul(2, Pi), ConstI), 5)), Neg(Exp(Div(Mul(Pi, ConstI), 5))), Exp(Div(Mul(Mul(4, Pi), ConstI), 5)), Neg(Exp(Div(Mul(Mul(3, Pi), ConstI), 5))), 1), Tuple(0, 1, Exp(Div(Mul(Mul(2, Pi), ConstI), 5)), Neg(Exp(Div(Mul(Pi, ConstI), 5))), Exp(Div(Mul(Mul(4, Pi), ConstI), 5)), Neg(Exp(Div(Mul(Mul(3, Pi), ConstI), 5))), Neg(Exp(Div(Mul(Mul(3, Pi), ConstI), 5))), Exp(Div(Mul(Mul(4, Pi), ConstI), 5)), Neg(Exp(Div(Mul(Pi, ConstI), 5))), Exp(Div(Mul(Mul(2, Pi), ConstI), 5)), 1), Tuple(0, 1, Exp(Div(Mul(Mul(4, Pi), ConstI), 5)), Exp(Div(Mul(Mul(2, Pi), ConstI), 5)), Neg(Exp(Div(Mul(Mul(3, Pi), ConstI), 5))), Neg(Exp(Div(Mul(Pi, ConstI), 5))), Neg(Exp(Div(Mul(Pi, ConstI), 5))), Neg(Exp(Div(Mul(Mul(3, Pi), ConstI), 5))), Exp(Div(Mul(Mul(2, Pi), ConstI), 5)), Exp(Div(Mul(Mul(4, Pi), ConstI), 5)), 1), Tuple(0, 1, Neg(Exp(Div(Mul(Mul(4, Pi), ConstI), 5))), Exp(Div(Mul(Mul(2, Pi), ConstI), 5)), Neg(Exp(Div(Mul(Mul(3, Pi), ConstI), 5))), Neg(Exp(Div(Mul(Pi, ConstI), 5))), Exp(Div(Mul(Pi, ConstI), 5)), Exp(Div(Mul(Mul(3, Pi), ConstI), 5)), Neg(Exp(Div(Mul(Mul(2, Pi), ConstI), 5))), Exp(Div(Mul(Mul(4, Pi), ConstI), 5)), -1), Tuple(0, 1, Neg(Exp(Div(Mul(Mul(2, Pi), ConstI), 5))), Neg(Exp(Div(Mul(Pi, ConstI), 5))), Exp(Div(Mul(Mul(4, Pi), ConstI), 5)), Neg(Exp(Div(Mul(Mul(3, Pi), ConstI), 5))), Exp(Div(Mul(Mul(3, Pi), ConstI), 5)), Neg(Exp(Div(Mul(Mul(4, Pi), ConstI), 5))), Exp(Div(Mul(Pi, ConstI), 5)), Exp(Div(Mul(Mul(2, Pi), ConstI), 5)), -1), Tuple(0, 1, Exp(Div(Mul(Mul(3, Pi), ConstI), 5)), Exp(Div(Mul(Mul(4, Pi), ConstI), 5)), Neg(Exp(Div(Mul(Pi, ConstI), 5))), Exp(Div(Mul(Mul(2, Pi), ConstI), 5)), Neg(Exp(Div(Mul(Mul(2, Pi), ConstI), 5))), Exp(Div(Mul(Pi, ConstI), 5)), Neg(Exp(Div(Mul(Mul(4, Pi), ConstI), 5))), Neg(Exp(Div(Mul(Mul(3, Pi), ConstI), 5))), -1), Tuple(0, 1, Neg(Exp(Div(Mul(Pi, ConstI), 5))), Neg(Exp(Div(Mul(Mul(3, Pi), ConstI), 5))), Exp(Div(Mul(Mul(2, Pi), ConstI), 5)), Exp(Div(Mul(Mul(4, Pi), ConstI), 5)), Exp(Div(Mul(Mul(4, Pi), ConstI), 5)), Exp(Div(Mul(Mul(2, Pi), ConstI), 5)), Neg(Exp(Div(Mul(Mul(3, Pi), ConstI), 5))), Neg(Exp(Div(Mul(Pi, ConstI), 5))), 1), Tuple(0, 1, -1, 1, 1, 1, -1, -1, -1, 1, -1))))

5e1d3b

Table of

\chi_{12 \, . \, \ell}

$\ell$ \ $n$	1	5	7	11
1	1	1	1	1
5	1	-1	1	-1
7	1	1	-1	-1
11	1	-1	-1	1

Table data:

\left(\ell, n, y\right)

such that

\chi(n) = y\; \text{ where } \chi = \chi_{12 \, . \, \ell}

Definitions:

Fungrim symbol	Notation	Short description
`DirichletCharacter`	$\chi_{q \, . \, \ell}$	Dirichlet character

Source code for this entry:

Entry(ID("5e1d3b"),
    Description("Table of", DirichletCharacter(12, ell)),
    Table(TableRelation(Tuple(ell, n, y), Where(Equal(chi(n), y), Equal(chi, DirichletCharacter(12, ell)))), TableHeadings(Description(ell, "\", n), 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11), TableColumnHeadings(1, 5, 7, 11), List(Tuple(0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1), Tuple(0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1), Tuple(0, 1, 0, 0, 0, 1, 0, -1, 0, 0, 0, -1), Tuple(0, 1, 0, 0, 0, -1, 0, -1, 0, 0, 0, 1))))

e6deb7

\left|\sum_{n=0}^{N} \chi(n)\right| \le \varphi(q)

Assumptions:

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; \chi \in G_{q} \;\mathbin{\operatorname{and}}\; \chi \ne \chi_{q \, . \, 1}

TeX:

\left|\sum_{n=0}^{N} \chi(n)\right| \le \varphi(q)

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; \chi \in G_{q} \;\mathbin{\operatorname{and}}\; \chi \ne \chi_{q \, . \, 1}

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`Sum`	$\sum_{n} f(n)$	Sum
`Totient`	$\varphi(n)$	Euler totient function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`ZZ`	$\mathbb{Z}$	Integers
`DirichletGroup`	$G_{q}$	Dirichlet characters with given modulus
`DirichletCharacter`	$\chi_{q \, . \, \ell}$	Dirichlet character

Source code for this entry:

Entry(ID("e6deb7"),
    Formula(LessEqual(Abs(Sum(chi(n), For(n, 0, N))), Totient(q))),
    Variables(N, chi, q),
    Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(N, ZZ), Element(chi, DirichletGroup(q)), NotEqual(chi, DirichletCharacter(q, 1)))))

5df909

\left|\sum_{n=M}^{N} \chi(n)\right| \le \frac{\sqrt{q} \log(q)}{2 \log(2)} + 3 \sqrt{q}

Assumptions:

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; M \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; \chi \in G_{q} \;\mathbin{\operatorname{and}}\; \chi \ne \chi_{q \, . \, 1}

Pólya-Vinogradov inequality, explicit form

References:

E. Dobrowolski and K. S. Williams, An upper bound for the sum ... for a certain class of functions f, Proceedings of the American Mathematical Society, Vol. 114, No. 1 (Jan., 1992), pp. 29-35, http://doi.org/10.2307/2159779

TeX:

\left|\sum_{n=M}^{N} \chi(n)\right| \le \frac{\sqrt{q} \log(q)}{2 \log(2)} + 3 \sqrt{q}

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; M \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; \chi \in G_{q} \;\mathbin{\operatorname{and}}\; \chi \ne \chi_{q \, . \, 1}

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`Sum`	$\sum_{n} f(n)$	Sum
`Sqrt`	$\sqrt{z}$	Principal square root
`Log`	$\log(z)$	Natural logarithm
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`ZZ`	$\mathbb{Z}$	Integers
`DirichletGroup`	$G_{q}$	Dirichlet characters with given modulus
`DirichletCharacter`	$\chi_{q \, . \, \ell}$	Dirichlet character

Source code for this entry:

Entry(ID("5df909"),
    Formula(LessEqual(Abs(Sum(chi(n), For(n, M, N))), Add(Div(Mul(Sqrt(q), Log(q)), Mul(2, Log(2))), Mul(3, Sqrt(q))))),
    Variables(q, chi, M, N),
    Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(M, ZZ), Element(N, ZZ), Element(chi, DirichletGroup(q)), NotEqual(chi, DirichletCharacter(q, 1)))),
    Description("Pólya-Vinogradov inequality, explicit form"),
    References("E. Dobrowolski and K. S. Williams, An upper bound for the sum ... for a certain class of functions f, Proceedings of the American Mathematical Society, Vol. 114, No. 1 (Jan., 1992), pp. 29-35, http://doi.org/10.2307/2159779"))

604c7c

L\!\left(s, \chi\right) = \sum_{n=1}^{\infty} \frac{\chi(n)}{{n}^{s}}

Assumptions:

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G_{q} \;\mathbin{\operatorname{and}}\; s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(s) > 1

TeX:

L\!\left(s, \chi\right) = \sum_{n=1}^{\infty} \frac{\chi(n)}{{n}^{s}}

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G_{q} \;\mathbin{\operatorname{and}}\; s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(s) > 1

Definitions:

Fungrim symbol	Notation	Short description
`DirichletL`	$L\!\left(s, \chi\right)$	Dirichlet L-function
`Sum`	$\sum_{n} f(n)$	Sum
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`DirichletGroup`	$G_{q}$	Dirichlet characters with given modulus
`CC`	$\mathbb{C}$	Complex numbers
`Re`	$\operatorname{Re}(z)$	Real part

Source code for this entry:

Entry(ID("604c7c"),
    Formula(Equal(DirichletL(s, chi), Sum(Div(chi(n), Pow(n, s)), For(n, 1, Infinity)))),
    Variables(q, chi, s),
    Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(chi, DirichletGroup(q)), Element(s, CC), Greater(Re(s), 1))))

Definitions

Character evaluation

Principal characters

Character group

Primitive decomposition

Conrey numbering

Multiplicativity

Odd powers

Even powers

Orthogonality

Tables

Primitive and non-primitive characters

Character values

Bounds and inequalities

L-series