Dirichlet character

Given an integer k, one defines the residue class of an integer n as the set of all integers congruent to n modulo k: ${\displaystyle {\hat {n}}=\{x\mid x\equiv n\ ({\text{mod}}\ k)\}.}$ That is, the residue class ${\displaystyle {\hat {n}}}$ is the coset of n in the quotient ring Z/kZ.

The set of units modulo k forms an abelian group of order ${\displaystyle \varphi (k)}$, where group multiplication is given by ${\displaystyle {\widehat {mn}}={\hat {m}}{\hat {n}}}$ and ${\displaystyle \varphi }$ again denotes Euler's phi function. The identity in this group is the residue class ${\displaystyle {\hat {1}}}$ and the inverse of ${\displaystyle {\hat {m}}}$ is the residue class ${\displaystyle {\hat {n}}}$ where ${\displaystyle {\hat {m}}{\hat {n}}={\hat {1}}}$, i.e., ${\displaystyle mn\equiv 1\mod k}$. For example, for k=6, the set of units is ${\displaystyle \{{\hat {1}},{\hat {5}}\}}$ because 0, 2, 3, and 4 are not coprime to 6.

The character group of (Z/k)^* consists of the residue class characters. A residue class character θ on (Z/k)^* is primitive if there is no proper divisor d of k such that θ factors as a map (Z/k)^* → (Z/d)^* → C^*, where the first arrow is the natural "modding d " map.[5]

The definition of a Dirichlet character modulo k ensures that it restricts to a character of the unit group modulo k:[6] a group homomorphism ${\displaystyle \chi }$ from (Z/kZ)^* to the non-zero complex numbers

${\displaystyle \chi$ :(\mathbb {Z} /k\mathbb {Z} )^{*}\to \mathbb {C} ^{*}} ,

with values that are necessarily roots of unity since the units modulo k form a finite group. In the opposite direction, given a group homomorphism ${\displaystyle \chi }$ on the unit group modulo k, we can lift to a completely multiplicative function on integers relatively prime to k and then extend this function to all integers by defining it to be 0 on integers having a non-trivial factor in common with k. The resulting function will then be a Dirichlet character.[7]

The principal character ${\displaystyle \chi _{0}}$ modulo k has the properties[7]

${\displaystyle \chi _{0}(n)=1}$ if gcd(n, k) = 1 and

${\displaystyle \chi _{0}(n)=0}$ if gcd(n, k) > 1.

The associated character of the multiplicative group (Z/kZ)^* is the principal character which always takes the value 1.[8]

When k is 1, the principal character modulo k is equal to 1 at all integers. For k greater than 1, the principal character modulo k vanishes at integers having a non-trivial common factor with k and is 1 at other integers.

There are φ(n) Dirichlet characters modulo n.[7]

A Dirichlet character is a completely multiplicative function ${\displaystyle f:\mathbb {N} \rightarrow \mathbb {C} }$ that satisfies a linear recurrence relation: that is, if ${\displaystyle a_{1}f(n+b_{1})+\cdots +a_{k}f(n+b_{k})=0}$

for all positive integer ${\displaystyle n}$, where ${\displaystyle a_{1},\ldots ,a_{k}}$ are not all zero and ${\displaystyle b_{1},\ldots ,b_{k}}$ are distinct then ${\displaystyle f}$ is a Dirichlet character.

A Dirichlet character is a completely multiplicative function ${\displaystyle f:\mathbb {N} \rightarrow \mathbb {C} }$ satisfying the following three properties: a) ${\displaystyle f}$ takes only finitely many values; b) ${\displaystyle f}$ vanishes at only finitely many primes; c) there is an ${\displaystyle \alpha \in \mathbb {C} }$ for which the remainder

${\displaystyle \left|\sum _{n\leq x}f(n)-\alpha x\right|}$

is uniformly bounded, as ${\displaystyle x\rightarrow \infty }$. This equivalent definition of Dirichlet characters was conjectured by Chudakov[10] in 1956, and proved in 2017 by Klurman and Mangerel.[11]

There is ${\displaystyle \varphi (1)=1}$ character modulo 1:

χ \ n	0
${\displaystyle \chi _{0}(n)}$	1

Note that χ is wholly determined by χ(0) since 0 generates the group of units modulo 1.

This is the trivial character.

The Dirichlet L-series for ${\displaystyle \chi _{0}(n)}$ is the Riemann zeta function

${\displaystyle \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}={\frac {1}{1^{s}}}+{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}+{\frac {1}{4^{s}}}+\cdots }$.

There is ${\displaystyle \varphi (2)=1}$ character modulo 2:

χ \ n	0	1
${\displaystyle \chi _{0}(n)}$	0	1

Note that χ is wholly determined by χ(1) since 1 generates the group of units modulo 2.

The Dirichlet L-series for ${\displaystyle \chi _{0}(n)}$ is the Dirichlet lambda function (closely related to the Dirichlet eta function)

${\displaystyle L(\chi _{0},s)=(1-2^{-s})\zeta (s)\,}$

There are ${\displaystyle \varphi (3)=2}$ characters modulo 3:

χ \ n	0	1	2
${\displaystyle \chi _{0}(n)}$	0	1	1
${\displaystyle \chi _{1}(n)}$	0	1	−1

Note that χ is wholly determined by χ(2) since 2 generates the group of units modulo 3.

There are ${\displaystyle \varphi (4)=2}$ characters modulo 4:

χ \ n	0	1	2	3
${\displaystyle \chi _{0}(n)}$	0	1	0	1
${\displaystyle \chi _{1}(n)}$	0	1	0	−1

Note that χ is wholly determined by χ(3) since 3 generates the group of units modulo 4.

The Dirichlet L-series for ${\displaystyle \chi _{0}(n)}$ is the Dirichlet lambda function (closely related to the Dirichlet eta function)

${\displaystyle L(\chi _{0},s)=(1-2^{-s})\zeta (s)\,}$

where ${\displaystyle \zeta (s)}$ is the Riemann zeta-function. The L-series for ${\displaystyle \chi _{1}(n)}$ is the Dirichlet beta-function

${\displaystyle L(\chi _{1},s)=\beta (s).\,}$

There are ${\displaystyle \varphi (5)=4}$ characters modulo 5. In the table below, i is the imaginary unit.

χ \ n	0	1	2	3	4
${\displaystyle \chi _{0}(n)}$	0	1	1	1	1
${\displaystyle \chi _{1}(n)}$	0	1	i	−i	−1
${\displaystyle \chi _{2}(n)}$	0	1	−1	−1	1
${\displaystyle \chi _{3}(n)}$	0	1	−i	i	−1

Note that χ is wholly determined by χ(2) and χ(3) since 2 and 3 generate the group of units modulo 5.

There are ${\displaystyle \varphi (6)=2}$ characters modulo 6:

χ \ n	0	1	2	3	4	5
${\displaystyle \chi _{0}(n)}$	0	1	0	0	0	1
${\displaystyle \chi _{1}(n)}$	0	1	0	0	0	−1

Note that χ is wholly determined by χ(5) since 5 generates the group of units modulo 6.

There are ${\displaystyle \varphi (7)=6}$ characters modulo 7. In the table below, ${\displaystyle \omega =e^{i\pi /3}.}$

χ \ n	0	1	2	3	4	5	6
${\displaystyle \chi _{0}(n)}$	0	1	1	1	1	1	1
${\displaystyle \chi _{1}(n)}$	0	1	ω2	ω	ω	ω2	1
${\displaystyle \chi _{2}(n)}$	0	1	ω	ω2	ω2	ω	1
${\displaystyle \chi _{3}(n)}$	0	1	1	−1	1	−1	−1
${\displaystyle \chi _{4}(n)}$	0	1	ω2	−ω	ω	−ω2	−1
${\displaystyle \chi _{5}(n)}$	0	1	ω	−ω2	ω2	−ω	−1

Note that χ is wholly determined by χ(3) since 3 generates the group of units modulo 7.

There are ${\displaystyle \varphi (8)=4}$ characters modulo 8.

χ \ n	0	1	2	3	4	5	6	7
${\displaystyle \chi _{0}(n)}$	0	1	0	1	0	1	0	1
${\displaystyle \chi _{1}(n)}$	0	1	0	1	0	−1	0	−1
${\displaystyle \chi _{2}(n)}$	0	1	0	−1	0	1	0	−1
${\displaystyle \chi _{3}(n)}$	0	1	0	−1	0	−1	0	1

Note that χ is wholly determined by χ(3) and χ(5) since 3 and 5 generate the group of units modulo 8.

There are ${\displaystyle \varphi (9)=6}$ characters modulo 9. In the table below, ${\displaystyle \omega =e^{i\pi /3}.}$

χ \ n	0	1	2	3	4	5	6	7	8
${\displaystyle \chi _{0}(n)}$	0	1	1	0	1	1	0	1	1
${\displaystyle \chi _{1}(n)}$	0	1	ω	0	ω2	−ω2	0	−ω	−1
${\displaystyle \chi _{2}(n)}$	0	1	ω2	0	−ω	−ω	0	ω2	1
${\displaystyle \chi _{3}(n)}$	0	1	−1	0	1	−1	0	1	−1
${\displaystyle \chi _{4}(n)}$	0	1	−ω	0	ω2	ω2	0	−ω	1
${\displaystyle \chi _{5}(n)}$	0	1	−ω2	0	−ω	ω	0	ω2	−1

Note that χ is wholly determined by χ(2) since 2 generates the group of units modulo 9.

There are ${\displaystyle \varphi (10)=4}$ characters modulo 10. In the table below, i is the imaginary unit.

χ \ n	0	1	2	3	4	5	6	7	8	9
${\displaystyle \chi _{0}(n)}$	0	1	0	1	0	0	0	1	0	1
${\displaystyle \chi _{1}(n)}$	0	1	0	i	0	0	0	−i	0	−1
${\displaystyle \chi _{2}(n)}$	0	1	0	−1	0	0	0	−1	0	1
${\displaystyle \chi _{3}(n)}$	0	1	0	−i	0	0	0	i	0	−1

Note that χ is wholly determined by χ(3) since 3 generates the group of units modulo 10.

There are ${\displaystyle \varphi (11)=10}$ characters modulo 11. In the table below, ${\displaystyle \omega =e^{i\pi /5}.}$

χ \ n	0	1	2	3	4	5	6	7	8	9	10
${\displaystyle \chi _{0}(n)}$	0	1	1	1	1	1	1	1	1	1	1
${\displaystyle \chi _{1}(n)}$	0	1	ω	ω3	ω2	ω4	ω4	ω2	ω3	ω	1
${\displaystyle \chi _{2}(n)}$	0	1	ω2	ω	ω4	ω3	ω3	ω4	ω	ω2	1
${\displaystyle \chi _{3}(n)}$	0	1	ω3	ω4	ω	ω2	ω2	ω	ω4	ω3	1
${\displaystyle \chi _{4}(n)}$	0	1	ω4	ω2	ω3	ω	ω	ω3	ω2	ω4	1
${\displaystyle \chi _{5}(n)}$	0	1	−1	1	1	1	−1	−1	−1	1	−1
${\displaystyle \chi _{6}(n)}$	0	1	−ω	ω3	ω2	ω4	−ω4	−ω2	−ω3	ω	−1
${\displaystyle \chi _{7}(n)}$	0	1	−ω2	ω	ω4	ω3	−ω3	−ω4	−ω	ω2	−1
${\displaystyle \chi _{8}(n)}$	0	1	−ω3	ω4	ω	ω2	−ω2	−ω	−ω4	ω3	−1
${\displaystyle \chi _{9}(n)}$	0	1	−ω4	ω2	ω3	ω	−ω	−ω3	−ω2	ω4	−1

Note that χ is wholly determined by χ(2) since 2 generates the group of units modulo 11.

There are ${\displaystyle \varphi (12)=4}$ characters modulo 12.

χ \ n	0	1	2	3	4	5	6	7	8	9	10	11
${\displaystyle \chi _{0}(n)}$	0	1	0	0	0	1	0	1	0	0	0	1
${\displaystyle \chi _{1}(n)}$	0	1	0	0	0	1	0	−1	0	0	0	−1
${\displaystyle \chi _{2}(n)}$	0	1	0	0	0	−1	0	1	0	0	0	−1
${\displaystyle \chi _{3}(n)}$	0	1	0	0	0	−1	0	−1	0	0	0	1

Note that χ is wholly determined by χ(5) and χ(7) since 5 and 7 generate the group of units modulo 12.