In analytic number theory and related branches of mathematics, a complex-valued arithmetic function is a Dirichlet character of modulus (where is a positive integer) if for all integers and :[1]
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 are denoted where the index is described in the section the group of characters below. In this labeling, denotes an unspecified character and denotes the principal character mod .
Relation to group characters
The word "character" is used several ways in mathematics. In this section it refers to a homomorphism from a group (written multiplicatively) to the multiplicative group of the field of complex numbers:
The set of characters is denoted If the product of two characters is defined by pointwise multiplication the identity by the trivial character and the inverse by complex inversion then becomes an abelian group.[7]
The elements of the finite abelian group are the residue classes where
A group character can be extended to a Dirichlet character by defining
and conversely, a Dirichlet character mod defines a group character on
Paraphrasing Davenport[10] 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 property 2) says so it can be canceled from both sides of :
The complex conjugate of a root of unity is also its inverse (see here for details), so for
( also obviously satisfies 1-3).
Thus for all integers
in other words .
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 have different structures depending on whether is a power of 2, a power of an odd prime, or the product of prime powers.[13]
Powers of odd primes
If is an odd number is cyclic of order ; a generator is called a primitive root mod .[14] Let be a primitive root and for define the function (the index of ) by
For if and only if Since
is determined by its value at
Let be a primitive -th root of unity. From property 7) above the possible values of are These distinct values give rise to Dirichlet characters mod For define as
Then for and all and
showing that is a character and
which gives an explicit isomorphism
Examples m = 3, 5, 7, 9
2 is a primitive root mod 3. ()
so the values of are
.
The nonzero values of the characters mod 3 are
2 is a primitive root mod 5. ()
so the values of are
.
The nonzero values of the characters mod 5 are
3 is a primitive root mod 7. ()
so the values of are
.
The nonzero values of the characters mod 7 are ()
.
2 is a primitive root mod 9. ()
so the values of are
.
The nonzero values of the characters mod 9 are ()
.
Powers of 2
is the trivial group with one element. 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 and their negatives are the units [15] For example
Let ; then is the direct product of a cyclic group of order 2 (generated by −1) and a cyclic group of order (generated by 5). For odd numbers define the functions and by
For odd and if and only if and For odd the value of is determined by the values of and
Let be a primitive -th root of unity. The possible values of are These distinct values give rise to Dirichlet characters mod For odd define by
Then for odd and and all and
showing that is a character and
showing that
Examples m = 2, 4, 8, 16
The only character mod 2 is the principal character .
−1 is a primitive root mod 4 ()
The nonzero values of the characters mod 4 are
−1 is and 5 generate the units mod 8 ()
.
The nonzero values of the characters mod 8 are
−1 and 5 generate the units mod 16 ()
.
The nonzero values of the characters mod 16 are
.
Products of prime powers
Let be the factorization of into prime powers. The group of units mod is isomorphic to the direct product of the groups mod the :[16]
This means that 1) there is a one-to-one correspondence between and -tuples where and 2) multiplication mod corresponds to coordinate-wise multiplication of -tuples: corresponds to where
There are Dirichlet characters mod They are denoted by where is equivalent to The identity is an isomorphism [22]
Each character mod has a unique factorization as the product of characters mod the prime powers dividing :
May 05, 2023
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In analytic number theory and related branches of mathematics a complex valued arithmetic function x Z C displaystyle chi mathbb Z rightarrow mathbb C is a Dirichlet character of modulus m displaystyle m where m displaystyle m is a positive integer if for all integers a displaystyle a and b displaystyle b 1 1 x a b x a x b displaystyle chi ab chi a chi b i e x displaystyle chi is completely multiplicative 2 x a 0 if gcd a m gt 1 0 if gcd a m 1 displaystyle chi a begin cases 0 amp text if gcd a m gt 1 neq 0 amp text if gcd a m 1 end cases gcd is the greatest common divisor 3 x a m x a displaystyle chi a m chi a i e x displaystyle chi is periodic with period m displaystyle m The simplest possible character called the principal character usually denoted x 0 displaystyle chi 0 see Notation below exists for all moduli 2 x 0 a 0 if gcd a m gt 1 1 if gcd a m 1 displaystyle chi 0 a begin cases 0 amp text if gcd a m gt 1 1 amp text if gcd a m 1 end cases The German mathematician Peter Gustav Lejeune Dirichlet for whom the character is named introduced these functions in his 1837 paper on primes in arithmetic progressions 3 4 Contents 1 Notation 2 Relation to group characters 3 Elementary facts 4 The group of characters 4 1 Powers of odd primes 4 1 1 Examples m 3 5 7 9 4 2 Powers of 2 4 2 1 Examples m 2 4 8 16 4 3 Products of prime powers 4 3 1 Examples m 15 24 40 4 4 Summary 5 Orthogonality 6 Classification of characters 6 1 Conductor Primitive and induced characters 6 2 Parity 6 3 Order 6 4 Real characters 6 4 1 Principal 6 4 2 Primitive 6 4 3 Imprimitive 7 Applications 7 1 L functions 7 2 Modular forms and functions 7 3 Gauss sum 7 4 Jacobi sum 7 5 Kloosterman sum 8 Sufficient conditions 8 1 From Davenport s book 8 2 Sarkozy s Condition 8 3 Chudakov s Condition 9 See also 10 Notes 11 References 12 External linksNotationϕ n displaystyle phi n is Euler s totient function z n displaystyle zeta n is a complex primitive n th root of unity z n n 1 displaystyle zeta n n 1 but z n 1 z n 2 1 z n n 1 1 displaystyle zeta n neq 1 zeta n 2 neq 1 zeta n n 1 neq 1 Z m Z displaystyle mathbb Z m mathbb Z times is the group of units mod m displaystyle m It has order ϕ m displaystyle phi m Z m Z displaystyle widehat mathbb Z m mathbb Z times is the group of Dirichlet characters mod m displaystyle m p p k displaystyle p p k etc are prime numbers m n displaystyle m n is a standard 5 abbreviation 6 for gcd m n displaystyle gcd m n x a x a x r a displaystyle 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 displaystyle m are denoted x m t a displaystyle chi m t a where the index t displaystyle t is described in the section the group of characters below In this labeling x m a displaystyle chi m a denotes an unspecified character and x m 1 a displaystyle chi m 1 a denotes the principal character mod m displaystyle m Relation to group charactersThe word character is used several ways in mathematics In this section it refers to a homomorphism from a group G displaystyle G written multiplicatively to the multiplicative group of the field of complex numbers h G C h g h h g h h h g 1 h g 1 displaystyle eta G rightarrow mathbb C times eta gh eta g eta h eta g 1 eta g 1 The set of characters is denoted G displaystyle widehat G If the product of two characters is defined by pointwise multiplication h 8 a h a 8 a displaystyle eta theta a eta a theta a the identity by the trivial character h 0 a 1 displaystyle eta 0 a 1 and the inverse by complex inversion h 1 a h a 1 displaystyle eta 1 a eta a 1 then G displaystyle widehat G becomes an abelian group 7 If A displaystyle A is a finite abelian group then 8 there are 1 an isomorphism A A displaystyle A cong widehat A and 2 the orthogonality relations 9 a A h a A if h h 0 0 if h h 0 displaystyle sum a in A eta a begin cases A amp text if eta eta 0 0 amp text if eta neq eta 0 end cases and h A h a A if a 1 0 if a 1 displaystyle sum eta in widehat A eta a begin cases A amp text if a 1 0 amp text if a neq 1 end cases The elements of the finite abelian group Z m Z displaystyle mathbb Z m mathbb Z times are the residue classes x y y x mod m displaystyle x y y equiv x pmod m where x m 1 displaystyle x m 1 A group character r Z m Z C displaystyle rho mathbb Z m mathbb Z times rightarrow mathbb C times can be extended to a Dirichlet character x Z C displaystyle chi mathbb Z rightarrow mathbb C by defining x a 0 if a Z m Z i e a m gt 1 r a if a Z m Z i e a m 1 displaystyle chi a begin cases 0 amp text if a not in mathbb Z m mathbb Z times amp text i e a m gt 1 rho a amp text if a in mathbb Z m mathbb Z times amp text i e a m 1 end cases and conversely a Dirichlet character mod m displaystyle m defines a group character on Z m Z displaystyle mathbb Z m mathbb Z times Paraphrasing Davenport 10 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 facts4 Since gcd 1 m 1 displaystyle gcd 1 m 1 property 2 says x 1 0 displaystyle chi 1 neq 0 so it can be canceled from both sides of x 1 x 1 x 1 1 x 1 displaystyle chi 1 chi 1 chi 1 times 1 chi 1 x 1 1 displaystyle chi 1 1 11 5 Property 3 is equivalent to if a b mod m displaystyle a equiv b pmod m then x a x b displaystyle chi a chi b 6 Property 1 implies that for any positive integer n displaystyle n x a n x a n displaystyle chi a n chi a n 7 Euler s theorem states that if a m 1 displaystyle a m 1 then a ϕ m 1 mod m displaystyle a phi m equiv 1 pmod m Therefore x a ϕ m x a ϕ m x 1 1 displaystyle chi a phi m chi a phi m chi 1 1 That is the nonzero values of x a displaystyle chi a are ϕ m displaystyle phi m th roots of unity x a 0 if gcd a m gt 1 z ϕ m r if gcd a m 1 displaystyle chi a begin cases 0 amp text if gcd a m gt 1 zeta phi m r amp text if gcd a m 1 end cases for some integer r displaystyle r which depends on x z displaystyle chi zeta and a displaystyle a This implies there are only a finite number of characters for a given modulus 8 If x displaystyle chi and x displaystyle chi are two characters for the same modulus so is their product x x displaystyle chi chi defined by pointwise multiplication x x a x a x a displaystyle chi chi a chi a chi a x x displaystyle chi chi obviously satisfies 1 3 12 The principal character is an identity x x 0 a x a x 0 a 0 0 x a if gcd a m gt 1 x a 1 x a if gcd a m 1 displaystyle chi chi 0 a chi a chi 0 a begin cases 0 times 0 amp chi a amp text if gcd a m gt 1 chi a times 1 amp chi a amp text if gcd a m 1 end cases 9 Let a 1 displaystyle a 1 denote the inverse of a displaystyle a in Z m Z displaystyle mathbb Z m mathbb Z times Then x a x a 1 x a a 1 x 1 1 displaystyle chi a chi a 1 chi aa 1 chi 1 1 so x a 1 x a 1 displaystyle 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 displaystyle a m 1 x a x a 1 x a 1 displaystyle overline chi a chi a 1 chi a 1 x displaystyle overline chi also obviously satisfies 1 3 Thus for all integers a displaystyle a x a x a 0 if gcd a m gt 1 1 if gcd a m 1 displaystyle chi a overline chi a begin cases 0 amp text if gcd a m gt 1 1 amp text if gcd a m 1 end cases in other words x x x 0 displaystyle 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 charactersThere are three different cases because the groups Z m Z displaystyle mathbb Z m mathbb Z times have different structures depending on whether m displaystyle m is a power of 2 a power of an odd prime or the product of prime powers 13 Powers of odd primes If q p k displaystyle q p k is an odd number Z q Z displaystyle mathbb Z q mathbb Z times is cyclic of order ϕ q displaystyle phi q a generator is called a primitive root mod q displaystyle q 14 Let g q displaystyle g q be a primitive root and for a q 1 displaystyle a q 1 define the function n q a displaystyle nu q a the index of a displaystyle a by a g q n q a mod q displaystyle a equiv g q nu q a pmod q 0 n q lt ϕ q displaystyle 0 leq nu q lt phi q For a b q 1 a b mod q displaystyle ab q 1 a equiv b pmod q if and only if n q a n q b displaystyle nu q a nu q b Since x a x g q n q a x g q n q a displaystyle chi a chi g q nu q a chi g q nu q a x displaystyle chi is determined by its value at g q displaystyle g q Let w q z ϕ q displaystyle omega q zeta phi q be a primitive ϕ q displaystyle phi q th root of unity From property 7 above the possible values of x g q displaystyle chi g q are w q w q 2 w q ϕ q 1 displaystyle omega q omega q 2 omega q phi q 1 These distinct values give rise to ϕ q displaystyle phi q Dirichlet characters mod q displaystyle q For r q 1 displaystyle r q 1 define x q r a displaystyle chi q r a as x q r a 0 if gcd a q gt 1 w q n q r n q a if gcd a q 1 displaystyle chi q r a begin cases 0 amp text if gcd a q gt 1 omega q nu q r nu q a amp text if gcd a q 1 end cases Then for r s q 1 displaystyle rs q 1 and all a displaystyle a and b displaystyle b x q r a x q r b x q r a b displaystyle chi q r a chi q r b chi q r ab showing that x q r displaystyle chi q r is a character and x q r a x q s a x q r s a displaystyle chi q r a chi q s a chi q rs a which gives an explicit isomorphism Z p k Z Z p k Z displaystyle 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 ϕ 3 2 displaystyle phi 3 2 2 1 2 2 2 2 0 1 mod 3 displaystyle 2 1 equiv 2 2 2 equiv 2 0 equiv 1 pmod 3 so the values of n 3 displaystyle nu 3 are a 1 2 n 3 a 0 1 displaystyle begin array c c c c c c c a amp 1 amp 2 hline nu 3 a amp 0 amp 1 end array The nonzero values of the characters mod 3 are 1 2 x 3 1 1 1 x 3 2 1 1 displaystyle begin array c c c c c c c amp 1 amp 2 hline chi 3 1 amp 1 amp 1 chi 3 2 amp 1 amp 1 end array 2 is a primitive root mod 5 ϕ 5 4 displaystyle phi 5 4 2 1 2 2 2 4 2 3 3 2 4 2 0 1 mod 5 displaystyle 2 1 equiv 2 2 2 equiv 4 2 3 equiv 3 2 4 equiv 2 0 equiv 1 pmod 5 so the values of n 5 displaystyle nu 5 are a 1 2 3 4 n 5 a 0 1 3 2 displaystyle begin array c c c c c c c a amp 1 amp 2 amp 3 amp 4 hline nu 5 a amp 0 amp 1 amp 3 amp 2 end array The nonzero values of the characters mod 5 are 1 2 3 4 x 5 1 1 1 1 1 x 5 2 1 i i 1 x 5 3 1 i i 1 x 5 4 1 1 1 1 displaystyle begin array c c c c c c c amp 1 amp 2 amp 3 amp 4 hline chi 5 1 amp 1 amp 1 amp 1 amp 1 chi 5 2 amp 1 amp i amp i amp 1 chi 5 3 amp 1 amp i amp i amp 1 chi 5 4 amp 1 amp 1 amp 1 amp 1 end array 3 is a primitive root mod 7 ϕ 7 6 displaystyle phi 7 6 3 1 3 3 2 2 3 3 6 3 4 4 3 5 5 3 6 3 0 1 mod 7 displaystyle 3 1 equiv 3 3 2 equiv 2 3 3 equiv 6 3 4 equiv 4 3 5 equiv 5 3 6 equiv 3 0 equiv 1 pmod 7 so the values of n 7 displaystyle nu 7 are a 1 2 3 4 5 6 n 7 a 0 2 1 4 5 3 displaystyle begin array c c c c c c c a amp 1 amp 2 amp 3 amp 4 amp 5 amp 6 hline nu 7 a amp 0 amp 2 amp 1 amp 4 amp 5 amp 3 end array The nonzero values of the characters mod 7 are w z 6 w 3 1 displaystyle omega zeta 6 omega 3 1 1 2 3 4 5 6 x 7 1 1 1 1 1 1 1 x 7 2 1 w w 2 w 2 w 1 x 7 3 1 w 2 w w w 2 1 x 7 4 1 w 2 w w w 2 1 x 7 5 1 w w 2 w 2 w 1 x 7 6 1 1 1 1 1 1 displaystyle begin array c c c c c c c amp 1 amp 2 amp 3 amp 4 amp 5 amp 6 hline chi 7 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 chi 7 2 amp 1 amp omega amp omega 2 amp omega 2 amp omega amp 1 chi 7 3 amp 1 amp omega 2 amp omega amp omega amp omega 2 amp 1 chi 7 4 amp 1 amp omega 2 amp omega amp omega amp omega 2 amp 1 chi 7 5 amp 1 amp omega amp omega 2 amp omega 2 amp omega amp 1 chi 7 6 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 end array 2 is a primitive root mod 9 ϕ 9 6 displaystyle phi 9 6 2 1 2 2 2 4 2 3 8 2 4 7 2 5 5 2 6 2 0 1 mod 9 displaystyle 2 1 equiv 2 2 2 equiv 4 2 3 equiv 8 2 4 equiv 7 2 5 equiv 5 2 6 equiv 2 0 equiv 1 pmod 9 so the values of n 9 displaystyle nu 9 are a 1 2 4 5 7 8 n 9 a 0 1 2 5 4 3 displaystyle begin array c c c c c c c a amp 1 amp 2 amp 4 amp 5 amp 7 amp 8 hline nu 9 a amp 0 amp 1 amp 2 amp 5 amp 4 amp 3 end array The nonzero values of the characters mod 9 are w z 6 w 3 1 displaystyle omega zeta 6 omega 3 1 1 2 4 5 7 8 x 9 1 1 1 1 1 1 1 x 9 2 1 w w 2 w 2 w 1 x 9 4 1 w 2 w w w 2 1 x 9 5 1 w 2 w w w 2 1 x 9 7 1 w w 2 w 2 w 1 x 9 8 1 1 1 1 1 1 displaystyle begin array c c c c c c c amp 1 amp 2 amp 4 amp 5 amp 7 amp 8 hline chi 9 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 chi 9 2 amp 1 amp omega amp omega 2 amp omega 2 amp omega amp 1 chi 9 4 amp 1 amp omega 2 amp omega amp omega amp omega 2 amp 1 chi 9 5 amp 1 amp omega 2 amp omega amp omega amp omega 2 amp 1 chi 9 7 amp 1 amp omega amp omega 2 amp omega 2 amp omega amp 1 chi 9 8 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 end array Powers of 2 Z 2 Z displaystyle mathbb Z 2 mathbb Z times is the trivial group with one element Z 4 Z displaystyle 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 1 mod 4 displaystyle equiv 1 pmod 4 and their negatives are the units 3 mod 4 displaystyle equiv 3 pmod 4 15 For example 5 1 5 5 2 5 0 1 mod 8 displaystyle 5 1 equiv 5 5 2 equiv 5 0 equiv 1 pmod 8 5 1 5 5 2 9 5 3 13 5 4 5 0 1 mod 16 displaystyle 5 1 equiv 5 5 2 equiv 9 5 3 equiv 13 5 4 equiv 5 0 equiv 1 pmod 16 5 1 5 5 2 25 5 3 29 5 4 17 5 5 21 5 6 9 5 7 13 5 8 5 0 1 mod 32 displaystyle 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 equiv 5 0 equiv 1 pmod 32 Let q 2 k k 3 displaystyle q 2 k k geq 3 then Z q Z displaystyle 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 ϕ q 2 displaystyle frac phi q 2 generated by 5 For odd numbers a displaystyle a define the functions n 0 displaystyle nu 0 and n q displaystyle nu q by a 1 n 0 a 5 n q a mod q displaystyle a equiv 1 nu 0 a 5 nu q a pmod q 0 n 0 lt 2 0 n q lt ϕ q 2 displaystyle 0 leq nu 0 lt 2 0 leq nu q lt frac phi q 2 For odd a displaystyle a and b a b mod q displaystyle b a equiv b pmod q if and only if n 0 a n 0 b displaystyle nu 0 a nu 0 b and n q a n q b displaystyle nu q a nu q b For odd a displaystyle a the value of x a displaystyle chi a is determined by the values of x 1 displaystyle chi 1 and x 5 displaystyle chi 5 Let w q z ϕ q 2 displaystyle omega q zeta frac phi q 2 be a primitive ϕ q 2 displaystyle frac phi q 2 th root of unity The possible values of x 1 n 0 a 5 n q a displaystyle chi 1 nu 0 a 5 nu q a are w q w q 2 w q ϕ q 2 1 displaystyle pm omega q pm omega q 2 pm omega q frac phi q 2 pm 1 These distinct values give rise to ϕ q displaystyle phi q Dirichlet characters mod q displaystyle q For odd r displaystyle r define x q r a displaystyle chi q r a by x q r a 0 if a is even 1 n 0 r n 0 a w q n q r n q a if a is odd displaystyle chi q r a begin cases 0 amp text if a text is even 1 nu 0 r nu 0 a omega q nu q r nu q a amp text if a text is odd end cases Then for odd r displaystyle r and s displaystyle s and all a displaystyle a and b displaystyle b x q r a x q r b x q r a b displaystyle chi q r a chi q r b chi q r ab showing that x q r displaystyle chi q r is a character and x q r a x q s a x q r s a displaystyle chi q r a chi q s a chi q rs a showing that Z 2 k Z Z 2 k Z displaystyle 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 x 2 1 displaystyle chi 2 1 1 is a primitive root mod 4 ϕ 4 2 displaystyle phi 4 2 a 1 3 n 0 a 0 1 displaystyle begin array a amp 1 amp 3 hline nu 0 a amp 0 amp 1 end array The nonzero values of the characters mod 4 are 1 3 x 4 1 1 1 x 4 3 1 1 displaystyle begin array c c c c c c c amp 1 amp 3 hline chi 4 1 amp 1 amp 1 chi 4 3 amp 1 amp 1 end array 1 is and 5 generate the units mod 8 ϕ 8 4 displaystyle phi 8 4 a 1 3 5 7 n 0 a 0 1 0 1 n 8 a 0 1 1 0 displaystyle begin array a amp 1 amp 3 amp 5 amp 7 hline nu 0 a amp 0 amp 1 amp 0 amp 1 nu 8 a amp 0 amp 1 amp 1 amp 0 end array The nonzero values of the characters mod 8 are 1 3 5 7 x 8 1 1 1 1 1 x 8 3 1 1 1 1 x 8 5 1 1 1 1 x 8 7 1 1 1 1 displaystyle begin array c c c c c c c amp 1 amp 3 amp 5 amp 7 hline chi 8 1 amp 1 amp 1 amp 1 amp 1 chi 8 3 amp 1 amp 1 amp 1 amp 1 chi 8 5 amp 1 amp 1 amp 1 amp 1 chi 8 7 amp 1 amp 1 amp 1 amp 1 end array 1 and 5 generate the units mod 16 ϕ 16 8 displaystyle phi 16 8 a 1 3 5 7 9 11 13 15 n 0 a 0 1 0 1 0 1 0 1 n 16 a 0 3 1 2 2 1 3 0 displaystyle begin array a amp 1 amp 3 amp 5 amp 7 amp 9 amp 11 amp 13 amp 15 hline nu 0 a amp 0 amp 1 amp 0 amp 1 amp 0 amp 1 amp 0 amp 1 nu 16 a amp 0 amp 3 amp 1 amp 2 amp 2 amp 1 amp 3 amp 0 end array The nonzero values of the characters mod 16 are 1 3 5 7 9 11 13 15 x 16 1 1 1 1 1 1 1 1 1 x 16 3 1 i i 1 1 i i 1 x 16 5 1 i i 1 1 i i 1 x 16 7 1 1 1 1 1 1 1 1 x 16 9 1 1 1 1 1 1 1 1 x 16 11 1 i i 1 1 i i 1 x 16 13 1 i i 1 1 i i 1 x 16 15 1 1 1 1 1 1 1 1 displaystyle begin array amp 1 amp 3 amp 5 amp 7 amp 9 amp 11 amp 13 amp 15 hline chi 16 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 chi 16 3 amp 1 amp i amp i amp 1 amp 1 amp i amp i amp 1 chi 16 5 amp 1 amp i amp i amp 1 amp 1 amp i amp i amp 1 chi 16 7 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 chi 16 9 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 chi 16 11 amp 1 amp i amp i amp 1 amp 1 amp i amp i amp 1 chi 16 13 amp 1 amp i amp i amp 1 amp 1 amp i amp i amp 1 chi 16 15 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 end array Products of prime powers Let m p 1 a 1 p 2 a 2 p k a k q 1 q 2 q k p 1 lt p 2 lt lt p k displaystyle m p 1 a 1 p 2 a 2 p k a k q 1 q 2 q k p 1 lt p 2 lt lt p k be the factorization of m displaystyle m into prime powers The group of units mod m displaystyle m is isomorphic to the direct product of the groups mod the q i displaystyle q i 16 Z m Z Z q 1 Z Z q 2 Z Z q k Z displaystyle mathbb Z m mathbb Z times cong mathbb Z q 1 mathbb Z times times mathbb Z q 2 mathbb Z times times times mathbb Z q k mathbb Z times This means that 1 there is a one to one correspondence between a Z m Z displaystyle a in mathbb Z m mathbb Z times and k displaystyle k tuples a 1 a 2 a k displaystyle a 1 a 2 a k where a i Z q i Z displaystyle a i in mathbb Z q i mathbb Z times and 2 multiplication mod m displaystyle m corresponds to coordinate wise multiplication of k displaystyle k tuples a b c mod m displaystyle ab equiv c pmod m corresponds to c 1 c 2 c k displaystyle c 1 c 2 c k where c i a i b i mod q i displaystyle c i equiv a i b i pmod q i The Chinese remainder theorem CRT implies that the a i displaystyle a i are simply a i a mod q i displaystyle a i equiv a pmod q i There are subgroups G i lt Z m Z displaystyle G i lt mathbb Z m mathbb Z times such that 17 G i Z q i Z displaystyle G i cong mathbb Z q i mathbb Z times and G i Z q i Z mod q i 1 mod q j j i displaystyle G i equiv begin cases mathbb Z q i mathbb Z times amp text mod q i 1 amp text mod q j j neq i end cases Then Z m Z G 1 G 2 G k displaystyle mathbb Z m mathbb Z times cong G 1 times G 2 times times G k and every a Z m Z displaystyle a in mathbb Z m mathbb Z times corresponds to a k displaystyle k tuple b 1 b 2 b k displaystyle b 1 b 2 b k where b i G i displaystyle b i in G i and b i a mod q i displaystyle b i equiv a pmod q i Every a Z m Z displaystyle a in mathbb Z m mathbb Z times can be uniquely factored as a b 1 b 2 b k displaystyle a b 1 b 2 b k 18 19 If x m displaystyle chi m is a character mod m displaystyle m on the subgroup G i displaystyle G i it must be identical to some x q i displaystyle chi q i mod q i displaystyle q i Then x m a x m b 1 b 2 x m b 1 x m b 2 x q 1 b 1 x q 2 b 2 displaystyle chi m a chi m b 1 b 2 chi m b 1 chi m b 2 chi q 1 b 1 chi q 2 b 2 showing that every character mod m displaystyle m is the product of characters mod the q i displaystyle q i For t m 1 displaystyle t m 1 define 20 x m t x q 1 t x q 2 t displaystyle chi m t chi q 1 t chi q 2 t Then for r s m 1 displaystyle rs m 1 and all a displaystyle a and b displaystyle b 21 x m r a x m r b x m r a b displaystyle chi m r a chi m r b chi m r ab showing that x m r displaystyle chi m r is a character and x m r a x m s a x m r s a displaystyle chi m r a chi m s a chi m rs a showing an isomorphism Z m Z Z m Z displaystyle widehat mathbb Z m mathbb Z times cong mathbb Z m mathbb Z times Examples m 15 24 40 Z 15 Z Z 3 Z Z 5 Z displaystyle 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 x 5 1 x 5 2 x 5 3 x 5 4 x 3 1 x 15 1 x 15 7 x 15 13 x 15 4 x 3 2 x 15 11 x 15 2 x 15 8 x 15 14 displaystyle begin array c c c c c c c amp chi 5 1 amp chi 5 2 amp chi 5 3 amp chi 5 4 hline chi 3 1 amp chi 15 1 amp chi 15 7 amp chi 15 13 amp chi 15 4 chi 3 2 amp chi 15 11 amp chi 15 2 amp chi 15 8 amp chi 15 14 end array The nonzero values of the characters mod 15 are 1 2 4 7 8 11 13 14 x 15 1 1 1 1 1 1 1 1 1 x 15 2 1 i 1 i i 1 i 1 x 15 4 1 1 1 1 1 1 1 1 x 15 7 1 i 1 i i 1 i 1 x 15 8 1 i 1 i i 1 i 1 x 15 11 1 1 1 1 1 1 1 1 x 15 13 1 i 1 i i 1 i 1 x 15 14 1 1 1 1 1 1 1 1 displaystyle begin array amp 1 amp 2 amp 4 amp 7 amp 8 amp 11 amp 13 amp 14 hline chi 15 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 chi 15 2 amp 1 amp i amp 1 amp i amp i amp 1 amp i amp 1 chi 15 4 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 chi 15 7 amp 1 amp i amp 1 amp i amp i amp 1 amp i amp 1 chi 15 8 amp 1 amp i amp 1 amp i amp i amp 1 amp i amp 1 chi 15 11 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 chi 15 13 amp 1 amp i amp 1 amp i amp i amp 1 amp i amp 1 chi 15 14 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 end array Z 24 Z Z 8 Z Z 3 Z displaystyle 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 x 8 1 x 8 3 x 8 5 x 8 7 x 3 1 x 24 1 x 24 19 x 24 13 x 24 7 x 3 2 x 24 17 x 24 11 x 24 5 x 24 23 displaystyle begin array c c c c c c c amp chi 8 1 amp chi 8 3 amp chi 8 5 amp chi 8 7 hline chi 3 1 amp chi 24 1 amp chi 24 19 amp chi 24 13 amp chi 24 7 chi 3 2 amp chi 24 17 amp chi 24 11 amp chi 24 5 amp chi 24 23 end array The nonzero values of the characters mod 24 are 1 5 7 11 13 17 19 23 x 24 1 1 1 1 1 1 1 1 1 x 24 5 1 1 1 1 1 1 1 1 x 24 7 1 1 1 1 1 1 1 1 x 24 11 1 1 1 1 1 1 1 1 x 24 13 1 1 1 1 1 1 1 1 x 24 17 1 1 1 1 1 1 1 1 x 24 19 1 1 1 1 1 1 1 1 x 24 23 1 1 1 1 1 1 1 1 displaystyle begin array amp 1 amp 5 amp 7 amp 11 amp 13 amp 17 amp 19 amp 23 hline chi 24 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 chi 24 5 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 chi 24 7 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 chi 24 11 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 chi 24 13 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 chi 24 17 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 chi 24 19 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 chi 24 23 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 end array Z 40 Z Z 8 Z Z 5 Z displaystyle 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 x 8 1 x 8 3 x 8 5 x 8 7 x 5 1 x 40 1 x 40 11 x 40 21 x 40 31 x 5 2 x 40 17 x 40 27 x 40 37 x 40 7 x 5 3 x 40 33 x 40 3 x 40 13 x 40 23 x 5 4 x 40 9 x 40 19 x 40 29 x 40 39 displaystyle begin array c c c c c c c amp chi 8 1 amp chi 8 3 amp chi 8 5 amp chi 8 7 hline chi 5 1 amp chi 40 1 amp chi 40 11 amp chi 40 21 amp chi 40 31 chi 5 2 amp chi 40 17 amp chi 40 27 amp chi 40 37 amp chi 40 7 chi 5 3 amp chi 40 33 amp chi 40 3 amp chi 40 13 amp chi 40 23 chi 5 4 amp chi 40 9 amp chi 40 19 amp chi 40 29 amp chi 40 39 end array The nonzero values of the characters mod 40 are 1 3 7 9 11 13 17 19 21 23 27 29 31 33 37 39 x 40 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 x 40 3 1 i i 1 1 i i 1 1 i i 1 1 i i 1 x 40 7 1 i i 1 1 i i 1 1 i i 1 1 i i 1 x 40 9 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 x 40 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 x 40 13 1 i i 1 1 i i 1 1 i i 1 1 i i 1 x 40 17 1 i i 1 1 i i 1 1 i i 1 1 i i 1 x 40 19 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 x 40 21 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 x 40 23 1 i i 1 1 i i 1 1 i i 1 1 i i 1 x 40 27 1 i i 1 1 i i 1 1 i i 1 1 i i 1 x 40 29 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 x 40 31 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 x 40 33 1 i i 1 1 i i 1 1 i i 1 1 i i 1 x 40 37 1 i i 1 1 i i 1 1 i i 1 1 i i 1 x 40 39 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 displaystyle begin array amp 1 amp 3 amp 7 amp 9 amp 11 amp 13 amp 17 amp 19 amp 21 amp 23 amp 27 amp 29 amp 31 amp 33 amp 37 amp 39 hline chi 40 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 chi 40 3 amp 1 amp i amp i amp 1 amp 1 amp i amp i amp 1 amp 1 amp i amp i amp 1 amp 1 amp i amp i amp 1 chi 40 7 amp 1 amp i amp i amp 1 amp 1 amp i amp i amp 1 amp 1 amp i amp i amp 1 amp 1 amp i amp i amp 1 chi 40 9 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 chi 40 11 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 chi 40 13 amp 1 amp i amp i amp 1 amp 1 amp i amp i amp 1 amp 1 amp i amp i amp 1 amp 1 amp i amp i amp 1 chi 40 17 amp 1 amp i amp i amp 1 amp 1 amp i amp i amp 1 amp 1 amp i amp i amp 1 amp 1 amp i amp i amp 1 chi 40 19 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 chi 40 21 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 chi 40 23 amp 1 amp i amp i amp 1 amp 1 amp i amp i amp 1 amp 1 amp i amp i amp 1 amp 1 amp i amp i amp 1 chi 40 27 amp 1 amp i amp i amp 1 amp 1 amp i amp i amp 1 amp 1 amp i amp i amp 1 amp 1 amp i amp i amp 1 chi 40 29 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 chi 40 31 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 chi 40 33 amp 1 amp i amp i amp 1 amp 1 amp i amp i amp 1 amp 1 amp i amp i amp 1 amp 1 amp i amp i amp 1 chi 40 37 amp 1 amp i amp i amp 1 amp 1 amp i amp i amp 1 amp 1 amp i amp i amp 1 amp 1 amp i amp i amp 1 chi 40 39 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 amp 1 end array Summary Let m p 1 k 1 p 2 k 2 q 1 q 2 p 1 lt p 2 lt displaystyle m p 1 k 1 p 2 k 2 q 1 q 2 p 1 lt p 2 lt be the factorization of m displaystyle m and assume r s m 1 displaystyle rs m 1 There are ϕ m displaystyle phi m Dirichlet characters mod m displaystyle m They are denoted by x m r displaystyle chi m r where x m r x m s displaystyle chi m r chi m s is equivalent to r s mod m displaystyle r equiv s pmod m The identity x m r a x m s a x m r s a displaystyle chi m r a chi m s a chi m rs a is an isomorphism Z m Z Z m Z displaystyle widehat mathbb Z m mathbb Z times cong mathbb Z m mathbb Z times 22 Each character mod m displaystyle m has a unique factorization as the product of characters mod the prime powers dividing m displaystyle m mat, wikipedia, wiki, book, books, library,