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Dirichlet characters

Table of contents: Definitions - Character evaluation - Principal characters - Character group - Primitive decomposition - Conrey numbering - Orthogonality - Tables - Bounds and inequalities - L-series

Definitions

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Symbol: DirichletCharacter χq(,)\chi_{q}(\ell, \cdot) Dirichlet character
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Symbol: DirichletGroup GqG_{q} Dirichlet characters with given modulus
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Symbol: PrimitiveDirichletCharacters GqprimitiveG_{q}^{\text{primitive}} Primitive Dirichlet characters with given modulus

Character evaluation

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χ ⁣(n)=χq(,n)   where χ=χq(,)\chi\!\left(n\right) = \chi_{q}(\ell, n)\; \text{ where } \chi = \chi_{q}(\ell, \cdot)
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χ ⁣(n){e2πik/r:kZ}{0}   where r=φ ⁣(q)\chi\!\left(n\right) \in \left\{ {e}^{2 \pi i k / r} : k \in \mathbb{Z} \right\} \cup \left\{0\right\}\; \text{ where } r = \varphi\!\left(q\right)
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χ ⁣(n+q)=χ ⁣(n)\chi\!\left(n + q\right) = \chi\!\left(n\right)
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χ ⁣(mn)=χ ⁣(m)χ ⁣(n)\chi\!\left(m n\right) = \chi\!\left(m\right) \chi\!\left(n\right)
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(χ ⁣(n)=0)    (gcd ⁣(n,q)1)\left(\chi\!\left(n\right) = 0\right) \iff \left(\gcd\!\left(n, q\right) \ne 1\right)
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χ ⁣(1)=1\chi\!\left(1\right) = 1
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χ ⁣(1){1,1}\chi\!\left(-1\right) \in \left\{1, -1\right\}
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χ ⁣(0)={1,q=10,q1\chi\!\left(0\right) = \begin{cases} 1, & q = 1\\0, & q \ne 1\\ \end{cases}

Principal characters

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χq(1,n)={1,gcd ⁣(n,q)=10,otherwise\chi_{q}(1, n) = \begin{cases} 1, & \gcd\!\left(n, q\right) = 1\\0, & \text{otherwise}\\ \end{cases}

Character group

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Gq={χq(,):{1,2,max ⁣(q,2)1}andgcd ⁣(,q)=1}G_{q} = \left\{ \chi_{q}(\ell, \cdot) : \ell \in \{1, 2, \ldots \max\!\left(q, 2\right) - 1\} \,\mathbin{\operatorname{and}}\, \gcd\!\left(\ell, q\right) = 1 \right\}
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Gqprimitive={χ:χGqand[for all d with d{1,2,q1}anddq,there exists a with a{0,1,q1}anda1(modd)andgcd ⁣(a,q)=1andχ ⁣(a)1]}G_{q}^{\text{primitive}} = \left\{ \chi : \chi \in G_{q} \,\mathbin{\operatorname{and}}\, \left[\text{for all } d \text{ with } d \in \{1, 2, \ldots q - 1\} \,\mathbin{\operatorname{and}}\, d \mid q, \,\, \text{there exists } a \text{ with } a \in \{0, 1, \ldots q - 1\} \,\mathbin{\operatorname{and}}\, a \equiv 1 \pmod {d} \,\mathbin{\operatorname{and}}\, \gcd\!\left(a, q\right) = 1 \,\mathbin{\operatorname{and}}\, \chi\!\left(a\right) \ne 1\right] \right\}
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#Gq=φ ⁣(q)\# G_{q} = \varphi\!\left(q\right)
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#Gqprimitive=dqφ ⁣(d)μ ⁣(qd)\# G_{q}^{\text{primitive}} = \sum_{d \mid q} \varphi\!\left(d\right) \mu\!\left(\frac{q}{d}\right)
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limNq=1N#Gq12N2=6π2\lim_{N \to \infty} \frac{\sum_{q=1}^{N} \# G_{q}}{\frac{1}{2} {N}^{2}} = \frac{6}{{\pi}^{2}}
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limNq=1N#Gqprimitiveq=1N#Gq=6π2\lim_{N \to \infty} \frac{\sum_{q=1}^{N} \# G_{q}^{\text{primitive}}}{\sum_{q=1}^{N} \# G_{q}} = \frac{6}{{\pi}^{2}}

Primitive decomposition

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there exists (d,χ0) with d{1,2,q}anddqandχ0Gdprimitiveandχ=χ0χ1   where χ1=χq(1,)\text{there exists } \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_{d}^{\text{primitive}} \,\mathbin{\operatorname{and}}\, \chi = {\chi}_{0} {\chi}_{1}\; \text{ where } {\chi}_{1} = \chi_{q}(1, \cdot)

Conrey numbering

Multiplicativity

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χq1q2(,)=χq1(modq1,)χq2(modq2,)\chi_{{q}_{1} {q}_{2}}(\ell, \cdot) = \chi_{{q}_{1}}(\ell \bmod {q}_{1}, \cdot) \chi_{{q}_{2}}(\ell \bmod {q}_{2}, \cdot)

Odd powers

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Symbol: ConreyGenerator gpg_{p} Conrey generator
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Symbol: DiscreteLog logb ⁣(x)modq\log_{b}\!\left(x\right) \bmod q Discrete logarithm
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gp=min{a:aZ1and#{akmodp:kZ0}=p1and#{akmodp2:kZ0}=p(p1)}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\}
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gp={10,p=404877,p=6692367337min(A),otherwise   where A={a:aZ1and#{akmodp:kZ0}=p1}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\}
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χq(,n)=exp ⁣(2πiabφ ⁣(q))   where q=pe,g=gp,a=logg ⁣()modq,b=logg ⁣(n)modq\chi_{q}(\ell, n) = \exp\!\left(\frac{2 \pi i a b}{\varphi\!\left(q\right)}\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

Even powers

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χ4(3,n)={1,n1(mod4)1,n3(mod4)0,otherwise\chi_{4}(3, n) = \begin{cases} 1, & n \equiv 1 \pmod {4}\\-1, & n \equiv 3 \pmod {4}\\0, & \text{otherwise}\\ \end{cases}
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χq(,n)=exp ⁣(2πi((1x)(1y)8+ab2e2))   where q=2e,L ⁣(k)={(1,log5 ⁣(k)modq),k{5imodq:iZ1}(1,log5 ⁣(k)modq),k{5imodq:iZ1},(x,a)=L ⁣(),(y,b)=L ⁣(n)\chi_{q}(\ell, n) = \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\!\left(k\right) = \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\!\left(\ell\right),\,\left(y, b\right) = L\!\left(n\right)

Orthogonality

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n=0q1χ ⁣(n)={φ ⁣(q),χ=χq(1,)0,otherwise\sum_{n=0}^{q - 1} \chi\!\left(n\right) = \begin{cases} \varphi\!\left(q\right), & \chi = \chi_{q}(1, \cdot)\\0, & \text{otherwise}\\ \end{cases}
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χGqχ ⁣(n)={φ ⁣(q),n1(modq)0,otherwise\sum_{\chi \in G_{q}} \chi\!\left(n\right) = \begin{cases} \varphi\!\left(q\right), & n \equiv 1 \pmod {q}\\0, & \text{otherwise}\\ \end{cases}
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n=0q1χ1 ⁣(n)χ2 ⁣(n)={φ ⁣(q),χ1=χ20,otherwise\sum_{n=0}^{q - 1} {\chi}_{1}\!\left(n\right) \overline{{\chi}_{2}\!\left(n\right)} = \begin{cases} \varphi\!\left(q\right), & {\chi}_{1} = {\chi}_{2}\\0, & \text{otherwise}\\ \end{cases}
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χGqχ ⁣(m)χ ⁣(n)={φ ⁣(q),nm(modq)andgcd ⁣(m,q)=10,otherwise\sum_{\chi \in G_{q}} \chi\!\left(m\right) \overline{\chi\!\left(n\right)} = \begin{cases} \varphi\!\left(q\right), & n \equiv m \pmod {q} \,\mathbin{\operatorname{and}}\, \gcd\!\left(m, q\right) = 1\\0, & \text{otherwise}\\ \end{cases}

Tables

Primitive and non-primitive characters

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Table of GqprimitiveG_{q}^{\text{primitive}} and GqGqprimitiveG_{q} \setminus G_{q}^{\text{primitive}} for 1q301 \le q \le 30

Character values

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Table of χ1(,)\chi_{1}(\ell, \cdot)
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Table of χ2(,)\chi_{2}(\ell, \cdot)
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Table of χ3(,)\chi_{3}(\ell, \cdot)
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Table of χ4(,)\chi_{4}(\ell, \cdot)
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Table of χ5(,)\chi_{5}(\ell, \cdot)
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Table of χ6(,)\chi_{6}(\ell, \cdot)
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Table of χ7(,)\chi_{7}(\ell, \cdot)
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Table of χ8(,)\chi_{8}(\ell, \cdot)
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Table of χ9(,)\chi_{9}(\ell, \cdot)
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Table of χ10(,)\chi_{10}(\ell, \cdot)
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Table of χ11(,)\chi_{11}(\ell, \cdot)
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Table of χ12(,)\chi_{12}(\ell, \cdot)

Bounds and inequalities

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n=0Nχ ⁣(n)φ ⁣(q)\left|\sum_{n=0}^{N} \chi\!\left(n\right)\right| \le \varphi\!\left(q\right)
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n=MNχ ⁣(n)qlog ⁣(q)2log ⁣(2)+3q\left|\sum_{n=M}^{N} \chi\!\left(n\right)\right| \le \frac{\sqrt{q} \log\!\left(q\right)}{2 \log\!\left(2\right)} + 3 \sqrt{q}

L-series

Related topics: Dirichlet L-functions

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

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-09-19 20:12:49.583742 UTC