# Dirichlet characters

## Definitions

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

## Character evaluation

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

## Principal characters

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

## Character group

$G_{q} = \left\{ \chi_{q}(\ell, \cdot) : \ell \in \{1, 2, \ldots \max\!\left(q, 2\right) - 1\} \,\mathbin{\operatorname{and}}\, \gcd\!\left(\ell, q\right) = 1 \right\}$
$G_{q}^{\text{primitive}} = \left\{ \chi : \chi \in G_{q} \,\mathbin{\operatorname{and}}\, \left[\text{for all } d \text{ with } d \in \{1, 2, \ldots q - 1\} \,\mathbin{\operatorname{and}}\, d \mid q, \,\, \text{there exists } a \text{ with } a \in \{0, 1, \ldots q - 1\} \,\mathbin{\operatorname{and}}\, a \equiv 1 \pmod {d} \,\mathbin{\operatorname{and}}\, \gcd\!\left(a, q\right) = 1 \,\mathbin{\operatorname{and}}\, \chi\!\left(a\right) \ne 1\right] \right\}$
$\# G_{q} = \varphi\!\left(q\right)$
$\# G_{q}^{\text{primitive}} = \sum_{d \mid q} \varphi\!\left(d\right) \mu\!\left(\frac{q}{d}\right)$
$\lim_{N \to \infty} \frac{\sum_{q=1}^{N} \# G_{q}}{\frac{1}{2} {N}^{2}} = \frac{6}{{\pi}^{2}}$
$\lim_{N \to \infty} \frac{\sum_{q=1}^{N} \# G_{q}^{\text{primitive}}}{\sum_{q=1}^{N} \# G_{q}} = \frac{6}{{\pi}^{2}}$

## Primitive decomposition

$\text{there exists } \left(d, {\chi}_{0}\right) \text{ with } d \in \{1, 2, \ldots q\} \,\mathbin{\operatorname{and}}\, d \mid q \,\mathbin{\operatorname{and}}\, {\chi}_{0} \in G_{d}^{\text{primitive}} \,\mathbin{\operatorname{and}}\, \chi = {\chi}_{0} {\chi}_{1}\; \text{ where } {\chi}_{1} = \chi_{q}(1, \cdot)$

## Conrey numbering

### Multiplicativity

$\chi_{{q}_{1} {q}_{2}}(\ell, \cdot) = \chi_{{q}_{1}}(\ell \bmod {q}_{1}, \cdot) \chi_{{q}_{2}}(\ell \bmod {q}_{2}, \cdot)$

### Odd powers

Symbol: ConreyGenerator $g_{p}$ Conrey generator
Symbol: DiscreteLog $\log_{b}\!\left(x\right) \bmod q$ Discrete logarithm
$g_{p} = \min \left\{ a : a \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, \# \left\{ {a}^{k} \bmod p : k \in \mathbb{Z}_{\ge 0} \right\} = p - 1 \,\mathbin{\operatorname{and}}\, \# \left\{ {a}^{k} \bmod {p}^{2} : k \in \mathbb{Z}_{\ge 0} \right\} = p \left(p - 1\right) \right\}$
$g_{p} = \begin{cases} 10, & p = 40487\\7, & p = 6692367337\\\min\left(A\right), & \text{otherwise}\\ \end{cases}\; \text{ where } A = \left\{ a : a \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, \# \left\{ {a}^{k} \bmod p : k \in \mathbb{Z}_{\ge 0} \right\} = p - 1 \right\}$
$\chi_{q}(\ell, n) = \exp\!\left(\frac{2 \pi i a b}{\varphi\!\left(q\right)}\right)\; \text{ where } q = {p}^{e},\,g = g_{p},\,a = \log_{g}\!\left(\ell\right) \bmod q,\,b = \log_{g}\!\left(n\right) \bmod q$

### Even powers

$\chi_{4}(3, n) = \begin{cases} 1, & n \equiv 1 \pmod {4}\\-1, & n \equiv 3 \pmod {4}\\0, & \text{otherwise}\\ \end{cases}$
$\chi_{q}(\ell, n) = \exp\!\left(2 \pi i \left(\frac{\left(1 - x\right) \left(1 - y\right)}{8} + \frac{a b}{{2}^{e - 2}}\right)\right)\; \text{ where } q = {2}^{e},\,L\!\left(k\right) = \begin{cases} \left(1, \log_{5}\!\left(k\right) \bmod q\right), & k \in \left\{ {5}^{i} \bmod q : i \in \mathbb{Z}_{\ge 1} \right\}\\\left(-1, \log_{5}\!\left(-k\right) \bmod q\right), & k \in \left\{ -{5}^{i} \bmod q : i \in \mathbb{Z}_{\ge 1} \right\}\\ \end{cases},\,\left(x, a\right) = L\!\left(\ell\right),\,\left(y, b\right) = L\!\left(n\right)$

## Orthogonality

$\sum_{n=0}^{q - 1} \chi\!\left(n\right) = \begin{cases} \varphi\!\left(q\right), & \chi = \chi_{q}(1, \cdot)\\0, & \text{otherwise}\\ \end{cases}$
$\sum_{\chi \in G_{q}} \chi\!\left(n\right) = \begin{cases} \varphi\!\left(q\right), & n \equiv 1 \pmod {q}\\0, & \text{otherwise}\\ \end{cases}$
$\sum_{n=0}^{q - 1} {\chi}_{1}\!\left(n\right) \overline{{\chi}_{2}\!\left(n\right)} = \begin{cases} \varphi\!\left(q\right), & {\chi}_{1} = {\chi}_{2}\\0, & \text{otherwise}\\ \end{cases}$
$\sum_{\chi \in G_{q}} \chi\!\left(m\right) \overline{\chi\!\left(n\right)} = \begin{cases} \varphi\!\left(q\right), & n \equiv m \pmod {q} \,\mathbin{\operatorname{and}}\, \gcd\!\left(m, q\right) = 1\\0, & \text{otherwise}\\ \end{cases}$

## Tables

### Primitive and non-primitive characters

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

### Character values

Table of $\chi_{1}(\ell, \cdot)$
Table of $\chi_{2}(\ell, \cdot)$
Table of $\chi_{3}(\ell, \cdot)$
Table of $\chi_{4}(\ell, \cdot)$
Table of $\chi_{5}(\ell, \cdot)$
Table of $\chi_{6}(\ell, \cdot)$
Table of $\chi_{7}(\ell, \cdot)$
Table of $\chi_{8}(\ell, \cdot)$
Table of $\chi_{9}(\ell, \cdot)$
Table of $\chi_{10}(\ell, \cdot)$
Table of $\chi_{11}(\ell, \cdot)$
Table of $\chi_{12}(\ell, \cdot)$

## Bounds and inequalities

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

## L-series

Related topics: Dirichlet L-functions

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

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