Overview
In analytic number theory and related branches of mathematics, a complex-valued arithmetic function \chi: \mathbb Z \rightarrow\mathbb C is a Dirichlet character of modulus m (where m is a positive integer) if for all integers a and b : :1. \chi(ab) = \chi(a)\chi(b); that is, \chi is completely multiplicative. :2. \chi(a) = 0 \iff \gcd(a,m)>1 . :3. \chi(a + m) = \chi(a) ; that is, \chi is periodic with period m . The simplest possible character, called the principal character and usually denoted \chi_0 , exists for all moduli: : \chi_0(a)= \begin cases 0 &\text if \gcd(a,m)>1\\ 1 &\text if \gcd(a,m)=1. \end cases Dirichlet characters were named after German mathematician Peter Gustav Lejeune Dirichlet, who introduced these functions in his 1837 paper on primes in arithmetic progressions. They are a prominent example of the general idea of a character in mathematics.
Notation
\phi(n) is Euler's totient function. \zeta_n is a complex primitive n-th root of unity: : \zeta_n^n=1, but \zeta_n\ne 1, \zeta_n^2\ne 1, ... \zeta_n^ n-1 \ne 1. (\mathbb Z /m\mathbb Z )^\times is the group of units mod m . It has order \phi(m). \widehat (\mathbb Z /m\mathbb Z )^\times is the group of Dirichlet characters mod m . p, p_k, etc. are prime numbers. (m,n) is a standard abbreviation for \gcd(m,n) \chi(a), \chi'(a), \chi_r(a), etc. are Dirichlet characters. (the lowercase Greek letter chi for "character") There is no standard notation for Dirichlet characters that includes the modulus. In many contexts (such as in the proof of Dirichlet's theorem) the modulus is fixed. In other contexts, such as this article, characters of different moduli appear. Where appropriate this article employs a variation of Conrey labeling (introduced by Brian Conrey and used by the LMFDB).
In this labeling characters for modulus m are denoted \chi_ m, t (a) where the index t is described in the section the group of characters below. In this labeling, \chi_ m,\_ (a) denotes an unspecified character and \chi_ m,1 (a) denotes the principal character mod m .
Relation to group characters
The word "character" is used several ways in mathematics. In this section it refers to a homomorphism from a group G (written multiplicatively) to the multiplicative group of the field of complex numbers: : \eta: G\rightarrow \mathbb C ^\times,\;\;\eta(gh)=\eta(g)\eta(h),\;\;\eta(g^ -1 )=\eta(g)^ -1 . The set of characters is denoted \widehat G . If the product of two characters is defined by pointwise multiplication \eta\theta(a)=\eta(a)\theta(a), the identity by the trivial character \eta_0(a)=1 and the inverse by complex inversion \eta^ -1 (a)=\eta(a)^ -1 then \widehat G becomes an abelian group.
If A is a finite abelian group then there is an isomorphism A\cong\widehat A , and the orthogonality relations: : \sum_ a\in A \eta(a)= \begin cases A &\text if \eta=\eta_0\\ 0&\text if \eta\ne\eta_0 \end cases and \sum_ \eta\in\widehat A \eta(a)= \begin cases A &\text if a=1\\ 0&\text if a\ne 1. \end cases The elements of the finite abelian group (\mathbb Z /m\mathbb Z )^\times are the residue classes a =\ x:x\equiv a\pmod m\ where (a,m)=1. A group character \rho:(\mathbb Z /m\mathbb Z )^\times\rightarrow \mathbb C ^\times can be extended to a Dirichlet character \chi:\mathbb Z \rightarrow \mathbb C by defining : \chi(a)= \begin cases 0 &\text if a \not\in(\mathbb Z /m\mathbb Z )^\times&\text i.e. (a,m)> 1\\ \rho( a )&\text if a \in(\mathbb Z /m\mathbb Z )^\times&\text i.e.
(a,m)= 1, \end cases and conversely, a Dirichlet character mod m defines a group character on (\mathbb Z /m\mathbb Z )^\times. Paraphrasing Davenport, Dirichlet characters can be regarded as a particular case of Abelian group characters. But this article follows Dirichlet in giving a direct and constructive account of them. This is partly for historical reasons, in that Dirichlet's work preceded by several decades the development of group theory, and partly for a mathematical reason, namely that the group in question has a simple and interesting structure which is obscured if one treats it as one treats the general Abelian group.
Elementary facts
4) Since \gcd(1,m)=1, property 2) says \chi(1)\ne 0 so it can be canceled from both sides of \chi(1)\chi(1)=\chi(1\times 1) =\chi(1) : : \chi(1)=1. 5) Property 3) is equivalent to :if a \equiv b \pmod m then \chi(a) =\chi(b). 6) Property 1) implies that, for any positive integer n : \chi(a^n)=\chi(a)^n. 7) Euler's theorem states that if \gcd(a,m)=1 then a^ \phi(m) \equiv 1 \pmod m . Therefore, : \chi(a)^ \phi(m) =\chi(a^ \phi(m) )=\chi(1)=1. That is, the nonzero values of \chi(a) are \phi(m) -th roots of unity: : \chi(a)= \begin cases 0 &\text if \gcd(a,m)>1\\ \zeta_ \phi(m) ^r&\text if \gcd(a,m)=1 \end cases for some integer r which depends on \chi, \zeta, and a . This implies there are only a finite number of characters for a given modulus.
8) If \chi and \chi' are two characters for the same modulus so is their product \chi\chi', defined by pointwise multiplication: : \chi\chi'(a) = \chi(a)\chi'(a) ( \chi\chi' obviously satisfies 1-3). The principal character is an identity: : \chi\chi_0(a)=\chi(a)\chi_0(a)= \begin cases 0 \times 0 &=\chi(a)&\text if \gcd(a,m)>1\\ \chi(a)\times 1&=\chi(a) &\text if \gcd(a,m)=1. \end cases 9) Let a^ -1 denote the inverse of a in (\mathbb Z /m\mathbb Z )^\times . Then : \chi(a)\chi(a^ -1 )=\chi(aa^ -1 )=\chi(1)=1, so \chi(a^ -1 )=\chi(a)^ -1 , which extends 6) to all integers. The complex conjugate of a root of unity is also its inverse (see here for details), so for (a,m)=1 : \overline \chi (a)=\chi(a)^ -1 =\chi(a^ -1 ). ( \overline\chi also obviously satisfies 1-3).
Thus for all integers a : \chi(a)\overline \chi (a)= \begin cases 0 &\text if \gcd(a,m)>1\\ 1 &\text if \gcd(a,m)=1 \end cases ; in other words \chi\overline \chi =\chi_0 . 10) The multiplication and identity defined in 8) and the inversion defined in 9) turn the set of Dirichlet characters for a given modulus into a finite abelian group.
The group of characters
There are three different cases because the groups (\mathbb Z /m\mathbb Z )^\times have different structures depending on whether m is a power of 2, a power of an odd prime, or the product of prime powers.
Powers of odd primes
If q=p^k is an odd number (\mathbb Z /q\mathbb Z )^\times is cyclic of order \phi(q) ; a generator is called a primitive root mod q . Let g_q be a primitive root and for (a,q)=1 define the function \nu_q(a) (the index of a ) by : a\equiv g_q^ \nu_q(a) \pmod q , : 0\le\nu_q For (ab,q)=1,\;\;a \equiv b\pmod q if and only if \nu_q(a)=\nu_q(b). Since : \chi(a)=\chi(g_q^ \nu_q(a) )=\chi(g_q)^ \nu_q(a) , \chi is determined by its value at g_q. Let \omega_q= \zeta_ \phi(q) be a primitive \phi(q) -th root of unity. From property 7) above the possible values of \chi(g_q) are \omega_q, \omega_q^2, ... \omega_q^ \phi(q) =1. These distinct values give rise to \phi(q) Dirichlet characters mod q.
For (r,q)=1 define \chi_ q,r (a) as : \chi_ q,r (a)= \begin cases 0 &\text if \gcd(a,q)>1\\ \omega_q^ \nu_q(r)\nu_q(a) &\text if \gcd(a,q)=1. \end cases Then for (rs,q)=1 and all a and b : \chi_ q,r (a)\chi_ q,r (b)=\chi_ q,r (ab), showing that \chi_ q,r is a character and : \chi_ q,r (a)\chi_ q,s (a)=\chi_ q,rs (a), which gives an explicit isomorphism \widehat (\mathbb Z /p^k\mathbb Z )^\times \cong(\mathbb Z /p^k\mathbb Z )^\times.
Examples m = 3, 5, 7, 9
2 is a primitive root mod 3. ( \phi(3)=2 ) : 2^1\equiv 2,\;2^2\equiv2^0\equiv 1\pmod 3 , so the values of \nu_3 are : \begin array a & 1 & 2 \\ \hline \nu_3(a) & 0 & 1\\ \end array . The nonzero values of the characters mod 3 are : \begin array & 1 & 2 \\ \hline \chi_ 3,1 & 1 & 1 \\ \chi_ 3,2 & 1 & -1 \\ \end array 2 is a primitive root mod 5. ( \phi(5)=4 ) : 2^1\equiv 2,\;2^2\equiv 4,\;2^3\equiv 3,\;2^4\equiv2^0\equiv 1\pmod 5 , so the values of \nu_5 are : \begin array a & 1 & 2 & 3 & 4 \\ \hline \nu_5(a) & 0 & 1 & 3 & 2 \\ \end array .
The nonzero values of the characters mod 5 are : \begin array & 1 & 2 & 3 & 4 \\ \hline \chi_ 5,1 & 1 & 1 & 1 & 1 \\ \chi_ 5,2 & 1 & i & -i & -1\\ \chi_ 5,3 & 1 & -i & i & -1\\ \chi_ 5,4 & 1 & -1 & -1 & 1\\ \end array 3 is a primitive root mod 7. ( \phi(7)=6 ) : 3^1\equiv 3,\;3^2\equiv 2,\;3^3\equiv 6,\;3^4\equiv 4,\;3^5\equiv 5,\;3^6\equiv3^0\equiv 1\pmod 7 , so the values of \nu_7 are : \begin array a & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \nu_7(a) & 0 & 2 & 1 & 4 & 5 & 3 \\ \end array .
The nonzero values of the characters mod 7 are ( \omega=\zeta_6, \;\;\omega^3=-1 ) : \begin array & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \chi_ 7,1 & 1 & 1 & 1 & 1 & 1 & 1 \\ \chi_ 7,2 & 1 & -\omega & \omega^2 & \omega^2 & -\omega & 1 \\ \chi_ 7,3 & 1 & \omega^2 & \omega & -\omega & -\omega^2 & -1 \\ \chi_ 7,4 & 1 & \omega^2 & -\omega & -\omega & \omega^2 & 1 \\ \chi_ 7,5 & 1 & -\omega & -\omega^2 & \omega^2 & \omega & -1 \\ \chi_ 7,6 & 1 & 1 & -1 & 1 & -1 & -1 \\ \end array . 2 is a primitive root mod 9. ( \phi(9)=6 ) : 2^1\equiv 2,\;2^2\equiv 4,\;2^3\equiv 8,\;2^4\equiv 7,\;2^5\equiv 5,\;2^6\equiv2^0\equiv 1\pmod 9 , so the values of \nu_9 are : \begin array a & 1 & 2 &4 & 5&7&8 \\ \hline \nu_9(a) & 0 & 1 & 2 & 5&4&3 \\ \end array .
The nonzero values of the characters mod 9 are ( \omega=\zeta_6, \;\;\omega^3=-1 ) : \begin array & 1 & 2 & 4 & 5 &7 & 8 \\ \hline \chi_ 9,1 & 1 & 1 & 1 & 1 & 1 & 1 \\ \chi_ 9,2 & 1 & \omega & \omega^2 & -\omega^2 & -\omega & -1 \\ \chi_ 9,4 & 1 & \omega^2 & -\omega & -\omega & \omega^2 & 1 \\ \chi_ 9,5 & 1 & -\omega^2 & -\omega & \omega & \omega^2 & -1 \\ \chi_ 9,7 & 1 & -\omega & \omega^2 & \omega^2 & -\omega & 1 \\ \chi_ 9,8 & 1 & -1 & 1 & -1 & 1 & -1 \\ \end array .
Powers of 2
(\mathbb Z /2\mathbb Z )^\times is the trivial group with one element. (\mathbb Z /4\mathbb Z )^\times is cyclic of order 2. For 8, 16, and higher powers of 2, there is no primitive root; the powers of 5 are the units \equiv 1\pmod 4 and their negatives are the units \equiv 3\pmod 4 . For example : 5^1\equiv 5,\;5^2\equiv5^0\equiv 1\pmod 8 : 5^1\equiv 5,\;5^2\equiv 9,\;5^3\equiv 13,\;5^4\equiv5^0\equiv 1\pmod 16 : 5^1\equiv 5,\;5^2\equiv 25,\;5^3\equiv 29,\;5^4\equiv 17,\;5^5\equiv 21,\;5^6\equiv 9,\;5^7\equiv 13,\;5^8\equiv5^0\equiv 1\pmod 32 . Let q=2^k, \;\;k\ge3 ; then (\mathbb Z /q\mathbb Z )^\times is the direct product of a cyclic group of order 2 (generated by −1) and a cyclic group of order \frac \phi(q) 2 (generated by 5).
For odd numbers a define the functions \nu_0 and \nu_q by : a\equiv(-1)^ \nu_0(a) 5^ \nu_q(a) \pmod q , : 0\le\nu_0 For odd a and b, \;\;a\equiv b\pmod q if and only if \nu_0(a)=\nu_0(b) and \nu_q(a)=\nu_q(b). For odd a the value of \chi(a) is determined by the values of \chi(-1) and \chi(5). Let \omega_q = \zeta_ \frac \phi(q) 2 be a primitive \frac \phi(q) 2 -th root of unity. The possible values of \chi((-1)^ \nu_0(a) 5^ \nu_q(a) ) are \pm\omega_q, \pm\omega_q^2, ... \pm\omega_q^ \frac \phi(q) 2 =\pm1. These distinct values give rise to \phi(q) Dirichlet characters mod q.
For odd r define \chi_ q,r (a) by : \chi_ q,r (a)= \begin cases 0 &\text if a\text is even \\ (-1)^ \nu_0(r)\nu_0(a) \omega_q^ \nu_q(r)\nu_q(a) &\text if a \text is odd . \end cases Then for odd r and s and all a and b : \chi_ q,r (a)\chi_ q,r (b)=\chi_ q,r (ab) showing that \chi_ q,r is a character and : \chi_ q,r (a)\chi_ q,s (a)=\chi_ q,rs (a) showing that \widehat (\mathbb Z /2^ k \mathbb Z )^\times \cong (\mathbb Z /2^ k \mathbb Z )^\times.
Examples m = 2, 4, 8, 16
The only character mod 2 is the principal character \chi_ 2,1 . −1 is a primitive root mod 4 ( \phi(4)=2 ) : \begin array a & 1 & 3 \\ \hline \nu_0(a) & 0 & 1 \\ \end array The nonzero values of the characters mod 4 are : \begin array & 1 & 3 \\ \hline \chi_ 4,1 & 1 & 1 \\ \chi_ 4,3 & 1 & -1 \\ \end array −1 is and 5 generate the units mod 8 ( \phi(8)=4 ) : \begin array a & 1 & 3 & 5 & 7 \\ \hline \nu_0(a) & 0 & 1 & 0 & 1 \\ \nu_8(a) & 0 & 1 & 1 & 0 \\ \end array .
The nonzero values of the characters mod 8 are : \begin array & 1 & 3 & 5 & 7 \\ \hline \chi_ 8,1 & 1 & 1 & 1 & 1 \\ \chi_ 8,3 & 1 & 1 & -1 & -1 \\ \chi_ 8,5 & 1 & -1 & -1 & 1 \\ \chi_ 8,7 & 1 & -1 & 1 & -1 \\ \end array −1 and 5 generate the units mod 16 ( \phi(16)=8 ) : \begin array a & 1 & 3 & 5 & 7 & 9 & 11 & 13 & 15 \\ \hline \nu_0(a) & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 \\ \nu_ 16 (a) & 0 & 3 & 1 & 2 & 2 & 1 & 3 & 0 \\ \end array .
The nonzero values of the characters mod 16 are : \begin array & 1 & 3 & 5 & 7 & 9 & 11 & 13 & 15 \\ \hline \chi_ 16,1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ \chi_ 16,3 & 1 & -i & -i & 1 & -1 & i & i & -1 \\ \chi_ 16,5 & 1 & -i & i & -1 & -1 & i & -i & 1 \\ \chi_ 16,7 & 1 & 1 & -1 & -1 & 1 & 1 & -1 & -1 \\ \chi_ 16,9 & 1 & -1 & -1 & 1 & 1 & -1 & -1 & 1 \\ \chi_ 16,11 & 1 & i & i & 1 & -1 & -i & -i & -1 \\ \chi_ 16,13 & 1 & i & -i & -1 & -1 & -i & i & 1 \\ \chi_ 16,15 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 \\ \end array .
Products of prime powers
Let m=p_1^ m_1 p_2^ m_2 \cdots p_k^ m_k = q_1q_2 \cdots q_k where p_1 be the factorization of m into prime powers. The group of units mod m is isomorphic to the direct product of the groups mod the q_i : : (\mathbb Z /m\mathbb Z )^\times \cong(\mathbb Z /q_1\mathbb Z )^\times \times(\mathbb Z /q_2\mathbb Z )^\times \times \dots \times(\mathbb Z /q_k\mathbb Z )^\times . This means that 1) there is a one-to-one correspondence between a\in (\mathbb Z /m\mathbb Z )^\times and k -tuples (a_1, a_2,\dots, a_k) where a_i\in(\mathbb Z /q_i\mathbb Z )^\times and 2) multiplication mod m corresponds to coordinate-wise multiplication of k -tuples: : ab\equiv c\pmod m corresponds to : (a_1,a_2,\dots,a_k)\times(b_1,b_2,\dots,b_k)=(c_1,c_2,\dots,c_k) where c_i\equiv a_ib_i\pmod q_i . The Chinese remainder theorem (CRT) implies that the a_i are simply a_i\equiv a\pmod q_i .
There are subgroups G_i such that : G_i\cong(\mathbb Z /q_i\mathbb Z )^\times and : G_i\equiv \begin cases (\mathbb Z /q_i\mathbb Z )^\times &\mod q_i\\ \ 1\ &\mod q_j, j\ne i. \end cases Then (\mathbb Z /m\mathbb Z )^\times \cong G_1\times G_2\times...\times G_k and every a\in (\mathbb Z /m\mathbb Z )^\times corresponds to a k -tuple (a_1, a_2,...a_k) where a_i\in G_i and a_i\equiv a\pmod q_i . Every a\in (\mathbb Z /m\mathbb Z )^\times can be uniquely factored as a =a_1a_2...a_k. If \chi_ m,\_ is a character mod m, on the subgroup G_i it must be identical to some \chi_ q_i,\_ mod q_i Then : \chi_ m,\_ (a)=\chi_ m,\_ (a_1a_2...)=\chi_ m,\_ (a_1)\chi_ m,\_ (a_2)...=\chi_ q_1,\_ (a_1)\chi_ q_2,\_ (a_2)..., showing that every character mod m is the product of characters mod the q_i . For (t,m)=1 define : \chi_ m,t =\chi_ q_1,t \chi_ q_2,t ...
Then for (rs,m)=1 and all a and b : \chi_ m,r (a)\chi_ m,r (b)=\chi_ m,r (ab), showing that \chi_ m,r is a character and : \chi_ m,r (a)\chi_ m,s (a)=\chi_ m,rs (a), showing an isomorphism \widehat (\mathbb Z /m\mathbb Z )^\times \cong(\mathbb Z /m\mathbb Z )^\times.
Examples m = 15, 24, 40
(\mathbb Z /15\mathbb Z )^\times\cong(\mathbb Z /3\mathbb Z )^\times\times(\mathbb Z /5\mathbb Z )^\times. The factorization of the characters mod 15 is : \begin array & \chi_ 5,1 & \chi_ 5,2 & \chi_ 5,3 & \chi_ 5,4 \\ \hline \chi_ 3,1 & \chi_ 15,1 & \chi_ 15,7 & \chi_ 15,13 & \chi_ 15,4 \\ \chi_ 3,2 & \chi_ 15,11 & \chi_ 15,2 & \chi_ 15,8 & \chi_ 15,14 \\ \end array The nonzero values of the characters mod 15 are : \begin array & 1 & 2 & 4 & 7 & 8 & 11 & 13 & 14 \\ \hline \chi_ 15,1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ \chi_ 15,2 & 1 & -i & -1 & i & i & -1 & -i & 1 \\ \chi_ 15,4 & 1 & -1 & 1 & -1 & -1 & 1 & -1 & 1 \\ \chi_ 15,7 & 1 & i & -1 & i & -i & 1 & -i & -1 \\ \chi_ 15,8 & 1 & i & -1 & -i & -i & -1 & i & 1 \\ \chi_ 15,11 & 1 & -1 & 1 & 1 & -1 & -1 & 1 & -1 \\ \chi_ 15,13 & 1 & -i & -1 & -i & i & 1 & i & -1 \\ \chi_ 15,14 & 1 & 1 & 1 & -1 & 1 & -1 & -1 & -1 \\ \end array .
(\mathbb Z /24\mathbb Z )^\times\cong(\mathbb Z /8\mathbb Z )^\times\times(\mathbb Z /3\mathbb Z )^\times. The factorization of the characters mod 24 is : \begin array & \chi_ 8,1 & \chi_ 8,3 & \chi_ 8,5 & \chi_ 8,7 \\ \hline \chi_ 3,1 & \chi_ 24,1 & \chi_ 24,19 & \chi_ 24,13 & \chi_ 24,7 \\ \chi_ 3,2 & \chi_ 24,17 & \chi_ 24,11 & \chi_ 24,5 & \chi_ 24,23 \\ \end array The nonzero values of the characters mod 24 are : \begin array & 1 & 5 & 7 & 11 & 13 & 17 & 19 & 23 \\ \hline \chi_ 24,1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ \chi_ 24,5 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 \\ \chi_ 24,7 & 1 & 1 & -1 & -1 & 1 & 1 & -1 & -1 \\ \chi_ 24,11 & 1 & 1 & -1 & -1 & -1 & -1 & 1 & 1 \\ \chi_ 24,13 & 1 & -1 & 1 & -1 & -1 & 1 & -1 & 1 \\ \chi_ 24,17 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 \\ \chi_ 24,19 & 1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 \\ \chi_ 24,23 & 1 & -1 & -1 & 1 & 1 & -1 & -1 & 1 \\ \end array .
(\mathbb Z /40\mathbb Z )^\times\cong(\mathbb Z /8\mathbb Z )^\times\times(\mathbb Z /5\mathbb Z )^\times.
The factorization of the characters mod 40 is : \begin array & \chi_ 8,1 & \chi_ 8,3 & \chi_ 8,5 & \chi_ 8,7 \\ \hline \chi_ 5,1 & \chi_ 40,1 & \chi_ 40,11 & \chi_ 40,21 & \chi_ 40,31 \\ \chi_ 5,2 & \chi_ 40,17 & \chi_ 40,27 & \chi_ 40,37 & \chi_ 40,7 \\ \chi_ 5,3 & \chi_ 40,33 & \chi_ 40,3 & \chi_ 40,13 & \chi_ 40,23 \\ \chi_ 5,4 & \chi_ 40,9 & \chi_ 40,19 & \chi_ 40,29 & \chi_ 40,39 \\ \end array The nonzero values of the characters mod 40 are : \begin array & 1 & 3 & 7 & 9 & 11 & 13 & 17 & 19 & 21 & 23 & 27 & 29 & 31 & 33 & 37 & 39 \\ \hline \chi_ 40,1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ \chi_ 40,3 & 1 & i & i & -1 & 1 & -i & -i & -1 & -1 & -i & -i & 1 & -1 & i & i & 1 \\ \chi_ 40,7 & 1 & i & -i & -1 & -1 & -i & i & 1 & 1 & i & -i & -1 & -1 & -i & i & 1 \\ \chi_ 40,9 & 1 & -1 & -1 & 1 & 1 & -1 & -1 & 1 & 1 & -1 & -1 & 1 & 1 & -1 & -1 & 1 \\ \chi_ 40,11 & 1 & 1 & -1 & 1 & 1 & -1 & 1 & 1 & -1 & -1 & 1 & -1 & -1 & 1 & -1 & -1 \\ \chi_ 40,13 & 1 & -i & -i & -1 & -1 & -i & -i & 1 & -1 & i & i & 1 & 1 & i & i & -1 \\ \chi_ 40,17 & 1 & -i & i & -1 & 1 & -i & i & -1 & 1 & -i & i & -1 & 1 & -i & i & -1 \\ \chi_ 40,19 & 1 & -1 & 1 & 1 & 1 & 1 & -1 & 1 & -1 & 1 & -1 & -1 & -1 & -1 & 1 & -1 \\ \chi_ 40,21 & 1 & -1 & 1 & 1 & -1 & -1 & 1 & -1 & -1 & 1 & -1 & -1 & 1 & 1 & -1 & 1 \\ \chi_ 40,23 & 1 & -i & i & -1 & -1 & i & -i & 1 & 1 & -i & i & -1 & -1 & i & -i & 1 \\ \chi_ 40,27 & 1 & -i & -i & -1 & 1 & i & i & -1 & -1 & i & i & 1 & -1 & -i & -i & 1 \\ \chi_ 40,29 & 1 & 1 & -1 & 1 & -1 & 1 & -1 & -1 & -1 & -1 & 1 & -1 & 1 & -1 & 1 & 1 \\ \chi_ 40,31 & 1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 & 1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 \\ \chi_ 40,33 & 1 & i & -i & -1 & 1 & i & -i & -1 & 1 & i & -i & -1 & 1 & i & -i & -1 \\ \chi_ 40,37 & 1 & i & i & -1 & -1 & i & i & 1 & -1 & -i & -i & 1 & 1 & -i & -i & -1 \\ \chi_ 40,39 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 \\ \end array .
Summary
Let m=p_1^ k_1 p_2^ k_2 \cdots = q_1q_2 \cdots , p_1 be the factorization of m and assume (rs,m)=1. There are \phi(m) Dirichlet characters mod m. They are denoted by \chi_ m,r , where \chi_ m,r =\chi_ m,s is equivalent to r\equiv s\pmod m . The identity \chi_ m,r (a)\chi_ m,s (a)=\chi_ m,rs (a)\; is an isomorphism \widehat (\mathbb Z /m\mathbb Z )^\times \cong(\mathbb Z /m\mathbb Z )^\times. Each character mod m has a unique factorization as the product of characters mod the prime powers dividing m : : \chi_ m,r =\chi_ q_1,r \chi_ q_2,r ... If m=m_1m_2, (m_1,m_2)=1 the product \chi_ m_1,r \chi_ m_2,s is a character \chi_ m,t where t is given by t\equiv r\pmod m_1 and t\equiv s\pmod m_2 . Also, \chi_ m,r (s)=\chi_ m,s (r)
Orthogonality
The two orthogonality relations are : \sum_ a\in(\mathbb Z /m\mathbb Z )^\times \chi(a)= \begin cases \phi(m)&\text if \;\chi=\chi_0\\ 0&\text if \;\chi\ne\chi_0 \end cases and \sum_ \chi\in\widehat (\mathbb Z /m\mathbb Z )^\times \chi(a)= \begin cases \phi(m)&\text if \;a\equiv 1\pmod m \\ 0&\text if \;a\not\equiv 1\pmod m . \end cases The relations can be written in the symmetric form : \sum_ a\in(\mathbb Z /m\mathbb Z )^\times \chi_ m,r (a)= \begin cases \phi(m)&\text if \;r\equiv 1\\ 0&\text if \;r\not\equiv 1 \end cases and \sum_ r\in(\mathbb Z /m\mathbb Z )^\times \chi_ m,r (a)= \begin cases \phi(m)&\text if \;a\equiv 1\\ 0&\text if \;a\not\equiv 1. \end cases The first relation is easy to prove: If \chi=\chi_0 there are \phi(m) non-zero summands each equal to 1.
If \chi\ne\chi_0 there is some a^*,\; (a^*,m)=1,\;\chi(a^*)\ne1. Then : \chi(a^*)\sum_ a\in(\mathbb Z /m\mathbb Z )^\times \chi(a)=\sum_ a \chi(a^*) \chi(a)=\sum_ a \chi(a^*a)=\sum_ a \chi(a), implying : (\chi(a^*)-1)\sum_ a \chi(a)=0. Dividing by the first factor gives \sum_ a \chi(a)=0, QED. The identity \chi_ m,r (s)=\chi_ m,s (r) for (rs,m)=1 shows that the relations are equivalent to each other. The second relation can be proven directly in the same way, but requires a lemma :Given a \not\equiv 1\pmod m ,\;(a,m)=1, there is a \chi^*,\; \chi^*(a)\ne1.
The second relation has an important corollary: if (a,m)=1, define the function : f_a(n)=\frac 1 \phi(m) \sum_ \chi \bar \chi (a) \chi(n). Then : f_a(n) = \frac 1 \phi(m) \sum_ \chi \chi(a^ -1 ) \chi(n) = \frac 1 \phi(m) \sum_ \chi \chi(a^ -1 n) = \begin cases 1, & n \equiv a \pmod m \\ 0, & n\not\equiv a\pmod m ,\end cases That is f_a=\mathbb 1 _ a the indicator function of the residue class a =\ x:\;x\equiv a \pmod m \ . It is basic in the proof of Dirichlet's theorem.
Conductor; Primitive and induced characters
Any character mod a prime power is also a character mod every larger power. For example, mod 16 : \begin array & 1 & 3 & 5 & 7 & 9 & 11 & 13 & 15 \\ \hline \chi_ 16,3 & 1 & -i & -i & 1 & -1 & i & i & -1 \\ \chi_ 16,9 & 1 & -1 & -1 & 1 & 1 & -1 & -1 & 1 \\ \chi_ 16,15 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 \\ \end array \chi_ 16,3 has period 16, but \chi_ 16,9 has period 8 and \chi_ 16,15 has period 4: \chi_ 16,9 =\chi_ 8,5 and \chi_ 16,15 =\chi_ 8,7 =\chi_ 4,3 . We say that a character \chi of modulus q has a quasiperiod of d if \chi(m)=\chi(n) for all m , n coprime to q satisfying m\equiv n mod d . For example, \chi_ 2,1 , the only Dirichlet character of modulus 2 , has a quasiperiod of 1 , but not a period of 1 (it has a period of 2 , though). The smallest positive integer for which \chi is quasiperiodic is the conductor of \chi .
So, for instance, \chi_ 2,1 has a conductor of 1 . The conductor of \chi_ 16,3 is 16, the conductor of \chi_ 16,9 is 8 and that of \chi_ 16,15 and \chi_ 8,7 is 4. If the modulus and conductor are equal the character is primitive, otherwise imprimitive. An imprimitive character is induced by the character for the smallest modulus: \chi_ 16,9 is induced from \chi_ 8,5 and \chi_ 16,15 and \chi_ 8,7 are induced from \chi_ 4,3 . A related phenomenon can happen with a character mod the product of primes; its nonzero values may be periodic with a smaller period. For example, mod 15, : \begin array & 1 & 2 &3 & 4 &5&6 & 7 & 8 &9&10 & 11&12 & 13 & 14 &15 \\ \hline \chi_ 15,8 & 1 & i &0 & -1 &0&0 & -i & -i &0&0 & -1 &0& i & 1 &0 \\ \chi_ 15,11 & 1 & -1 &0 & 1 &0&0 & 1 & -1 &0&0 & -1 &0& 1 & -1 &0\\ \chi_ 15,13 & 1 & -i &0 & -1 &0&0 & -i & i &0&0 & 1 &0 & i & -1 &0\\ \end array .
The nonzero values of \chi_ 15,8 have period 15, but those of \chi_ 15,11 have period 3 and those of \chi_ 15,13 have period 5. This is easier to see by juxtaposing them with characters mod 3 and 5: : \begin array & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 &15\\ \hline \chi_ 15,11 & 1 & -1 & 0 & 1 & 0 & 0 & 1 & -1 & 0 & 0 & -1 & 0 & 1 & -1 &0\\ \chi_ 3,2 & 1 & -1 & 0 & 1 & -1 & 0 & 1 & -1 & 0 & 1 & -1 & 0 & 1 & -1 &0\\ \hline \chi_ 15,13 & 1 & -i & 0 & -1 & 0 & 0 & -i & i & 0 & 0 & 1 & 0 & i & -1 &0\\ \chi_ 5,3 & 1 & -i & i & -1 & 0 & 1 & -i & i & -1 & 0 & 1 & -i & i & -1 &0\\ \end array .
If a character mod m=qr,\;\; (q,r)=1, \;\;q>1,\;\; r>1 is defined as : \chi_ m,\_ (a)= \begin cases 0&\text if \gcd(a,m)>1\\ \chi_ q,\_ (a)&\text if \gcd(a,m)=1 \end cases , or equivalently as \chi_ m,\_ = \chi_ q,\_ \chi_ r,1 , its nonzero values are determined by the character mod q and have period q . The smallest period of the nonzero values is the conductor of the character. For example, the conductor of \chi_ 15,8 is 15, the conductor of \chi_ 15,11 is 3, and that of \chi_ 15,13 is 5. As in the prime-power case, if the conductor equals the modulus the character is primitive, otherwise imprimitive. If imprimitive it is induced from the character with the smaller modulus. For example, \chi_ 15,11 is induced from \chi_ 3,2 and \chi_ 15,13 is induced from \chi_ 5,3 The principal character is not primitive.
The character \chi_ m,r =\chi_ q_1,r \chi_ q_2,r ... is primitive if and only if each of the factors is primitive. Primitive characters often simplify (or make possible) formulas in the theories of L-functions and modular forms.
Parity
\chi(a) is even if \chi(-1)=1 and is odd if \chi(-1)=-1. This distinction appears in the functional equation of the Dirichlet L-function.
Order
The order of a character is its order as an element of the group \widehat (\mathbb Z /m\mathbb Z )^\times , i.e. the smallest positive integer n such that \chi^n= \chi_0. Because of the isomorphism \widehat (\mathbb Z /m\mathbb Z )^\times \cong(\mathbb Z /m\mathbb Z )^\times the order of \chi_ m,r is the same as the order of r in (\mathbb Z /m\mathbb Z )^\times. The principal character has order 1; other real characters have order 2, and imaginary characters have order 3 or greater. By Lagrange's theorem the order of a character divides the order of \widehat (\mathbb Z /m\mathbb Z )^\times which is \phi(m)
Real characters
\chi(a) is real or quadratic if all of its values are real (they must be 0,\;\pm1 ); otherwise it is complex or imaginary. \chi is real if and only if \chi^2=\chi_0 ; \chi_ m,k is real if and only if k^2\equiv1\pmod m ; in particular, \chi_ m,-1 is real and non-principal. Dirichlet's original proof that L(1,\chi)\ne0 (which was only valid for prime moduli) took two different forms depending on whether \chi was real or not. His later proof, valid for all moduli, was based on his class number formula. Real characters are Kronecker symbols; for example, the principal character can be written \chi_ m,1 =\left(\frac m^2 \bullet \right) . The real characters in the examples are:
Principal
If m=p_1^ k_1 p_2^ k_2 ...,\;p_1 the principal character is \chi_ m,1 =\left(\frac p_1^2p_2^2... \bullet \right). \chi_ 16,1 =\chi_ 8,1 =\chi_ 4,1 =\chi_ 2,1 =\left(\frac 4 \bullet \right) \chi_ 9,1 =\chi_ 3,1 =\left(\frac 9 \bullet \right) \chi_ 5,1 =\left(\frac 25 \bullet \right) \chi_ 7,1 =\left(\frac 49 \bullet \right) \chi_ 15,1 =\left(\frac 225 \bullet \right) \chi_ 24,1 =\left(\frac 36 \bullet \right) \chi_ 40,1 =\left(\frac 100 \bullet \right)
Primitive
If the modulus is the absolute value of a fundamental discriminant there is a real primitive character (there are two if the modulus is a multiple of 8); otherwise if there are any primitive characters \chi_ 3,2 =\left(\frac -3 \bullet \right) \chi_ 4,3 =\left(\frac -4 \bullet \right) \chi_ 5,4 =\left(\frac 5 \bullet \right) \chi_ 7,6 =\left(\frac -7 \bullet \right) \chi_ 8,3 =\left(\frac -8 \bullet \right) \chi_ 8,5 =\left(\frac 8 \bullet \right) \chi_ 15,14 =\left(\frac -15 \bullet \right) \chi_ 24,5 =\left(\frac -24 \bullet \right) \chi_ 24,11 =\left(\frac 24 \bullet \right) \chi_ 40,19 =\left(\frac -40 \bullet \right) \chi_ 40,29 =\left(\frac 40 \bullet \right)
Imprimitive
\chi_ 8,7 =\chi_ 4,3 =\left(\frac -4 \bullet \right) \chi_ 9,8 =\chi_ 3,2 =\left(\frac -3 \bullet \right) \chi_ 15,4 =\chi_ 5,4 \chi_ 3,1 =\left(\frac 45 \bullet \right) \chi_ 15,11 =\chi_ 3,2 \chi_ 5,1 =\left(\frac -75 \bullet \right) \chi_ 16,7 =\chi_ 8,3 =\left(\frac -8 \bullet \right) \chi_ 16,9 =\chi_ 8,5 =\left(\frac 8 \bullet \right) \chi_ 16,15 =\chi_ 4,3 =\left(\frac -4 \bullet \right) \chi_ 24,7 =\chi_ 8,7 \chi_ 3,1 =\chi_ 4,3 \chi_ 3,1 =\left(\frac -36 \bullet \right) \chi_ 24,13 =\chi_ 8,5 \chi_ 3,1 =\left(\frac 72 \bullet \right) \chi_ 24,17 =\chi_ 3,2 \chi_ 8,1 =\left(\frac -12 \bullet \right) \chi_ 24,19 =\chi_ 8,3 \chi_ 3,1 =\left(\frac -72 \bullet \right) \chi_ 24,23 =\chi_ 8,7 \chi_ 3,2 =\chi_ 4,3 \chi_ 3,2 =\left(\frac 12 \bullet \right) \chi_ 40,9 =\chi_ 5,4 \chi_ 8,1 =\left(\frac 20 \bullet \right) \chi_ 40,11 =\chi_ 8,3 \chi_ 5,1 =\left(\frac -200 \bullet \right) \chi_ 40,21 =\chi_ 8,5 \chi_ 5,1 =\left(\frac 200 \bullet \right) \chi_ 40,31 =\chi_ 8,7 \chi_ 5,1 =\chi_ 4,3 \chi_ 5,1 =\left(\frac -100 \bullet \right) \chi_ 40,39 =\chi_ 8,7 \chi_ 5,4 =\chi_ 4,3 \chi_ 5,4 =\left(\frac -20 \bullet \right)
L-functions
The Dirichlet L-series for a character \chi is : L(s,\chi) = \sum_ n=1 ^\infty \frac \chi(n) n^s . This series converges absolutely for \mathfrak R (s) >1 . If the character is non-principal then furthermore it converges (but not absolutely) for \mathfrak R (s) >0 and it can be analytically continued to an entire function, defined and differentiable on the whole complex plane. If the character is principal then the series it converges only for \mathfrak R (s) >1 ; in this case, it can be analytically continued to a meromorphic function with simple pole at s = 1 . Dirichlet introduced the L -function along with the characters in his 1837 paper.
Modular forms and functions
Dirichlet characters appear several places in the theory of modular forms and functions. A typical example is Let \chi\in\widehat (\mathbb Z /M\mathbb Z )^\times and let \chi_1\in\widehat (\mathbb Z /N\mathbb Z )^\times be primitive. If : f(z)=\sum a_n z^n\in M_k(M,\chi) define : f_ \chi_1 (z)=\sum\chi_1(n)a_nz^n , Then : f_ \chi_1 (z)\in M_k(MN^2,\chi\chi_1^2) . If f is a cusp form so is f_ \chi_1 . See theta series of a Dirichlet character for another example.
Gauss sum
The Gauss sum of a Dirichlet character modulo is : G(\chi)=\sum_ a=1 ^N\chi(a)e^\frac 2\pi ia N . It appears in the functional equation of the Dirichlet L-function.
Jacobi sum
If \chi and \psi are Dirichlet characters mod a prime p their Jacobi sum is : J(\chi,\psi) = \sum_ a=2 ^ p-1 \chi(a) \psi(1 - a). Jacobi sums can be factored into products of Gauss sums.
Kloosterman sum
If \chi is a Dirichlet character mod q and \zeta = e^\frac 2\pi i q the Kloosterman sum K(a,b,\chi) is defined as : K(a,b,\chi)=\sum_ r\in (\mathbb Z /q\mathbb Z )^\times \chi(r)\zeta^ ar+\frac b r . If b=0 it is a Gauss sum.
Sufficient conditions
It is not necessary to establish the defining properties 1) – 3) to show that a function is a Dirichlet character.
From Davenport's book
If \Chi:\mathbb Z \rightarrow\mathbb C such that :1) \Chi(ab) = \Chi(a)\Chi(b), :2) \Chi(a + m) = \Chi(a) , :3) If \gcd(a,m)>1 then \Chi(a)=0 , but :4) \Chi(a) is not always 0, then \Chi(a) is one of the \phi(m) characters mod m
Sárközy's Condition
A Dirichlet character is a completely multiplicative function f: \mathbb N \rightarrow \mathbb C that satisfies a linear recurrence relation: that is, if a_1 f(n+b_1) + \cdots + a_kf(n+b_k) = 0 for all positive integers n , where a_1,\ldots,a_k are not all zero and b_1,\ldots,b_k are distinct then f is a Dirichlet character.
Chudakov's Condition
A Dirichlet character is a completely multiplicative function f: \mathbb N \rightarrow \mathbb C satisfying the following three properties: a) f takes only finitely many values; b) f vanishes at only finitely many primes; c) there is an \alpha \in \mathbb C for which the remainder \left \sum_ n \leq x f(n)- \alpha x\right is uniformly bounded, as x \rightarrow \infty . This equivalent definition of Dirichlet characters was conjectured by Chudakov in 1956, and proved in 2017 by Klurman and Mangerel.
Some notable special moduli
* 8, the smallest modulus whose Dirichlet characters need more than one generator. * 13, the smallest modulus whose Dirichlet characters contain numbers \alpha such that there is no primes p in \mathbb Z which are still primes in \mathbb Z \alpha . * 19, the smallest modulus whose Dirichlet characters contain numbers whose real and imaginary parts are not constructible numbers. * 24, the largest modulus whose Dirichlet characters are all real (the Dirichlet characters of the number n are all real if and only if n is divisor of 24). * 47, the smallest modulus whose Dirichlet characters contain numbers \alpha such that the class number h^- of the cyclotomic field \mathbb Q (\alpha) is greater than 1. * 120, the smallest modulus whose Dirichlet characters need more than three generators. * 149, the smallest modulus whose Dirichlet characters contain numbers \alpha such that the full class number h^- \cdot h^+ of the cyclotomic field \mathbb Q (\alpha) is not coprime to the smallest number such that \alpha^n=1 (related to irregular prime). * 240, the largest modulus whose Dirichlet characters are all Gaussian integers (the Dirichlet characters of the number n are all Gaussian integers if and only if n is divisor of 240). * 383, the smallest modulus whose Dirichlet characters contain numbers \alpha such that the class number h^+ of the cyclotomic field \mathbb Q (\alpha) is greater than 1. * 504, the largest modulus whose Dirichlet characters are all Eisenstein integers (the Dirichlet characters of the number n are all Eisenstein integers if and only if n is divisor of 504). * 840, the smallest modulus whose Dirichlet characters need more than four generators.
See also
* Character sum * Multiplicative group of integers modulo n * Primitive root modulo n * Multiplicative character
External links
* English translation of Dirichlet's 1837 paper on primes in arithmetic progressions * LMFDB Lists 30,397,486 Dirichlet characters of modulus up to 10,000 and their L-functions