HOME
The Info List - Dirichlet Character





In number theory, Dirichlet characters are certain arithmetic functions which arise from completely multiplicative characters on the units of

Z

/

k

Z

displaystyle mathbb Z /kmathbb Z

. Dirichlet characters are used to define Dirichlet L-functions, which are meromorphic functions with a variety of interesting analytic properties. If

χ

displaystyle chi

is a Dirichlet character, one defines its Dirichlet L-series by

L ( s , χ ) =

n = 1

χ ( n )

n

s

displaystyle L(s,chi )=sum _ n=1 ^ infty frac chi (n) n^ s

where s is a complex number with real part > 1. By analytic continuation, this function can be extended to a meromorphic function on the whole complex plane. Dirichlet L-functions are generalizations of the Riemann zeta-function and appear prominently in the generalized Riemann hypothesis. Dirichlet characters are named in honour of Peter Gustav Lejeune Dirichlet.

Contents

1 Axiomatic definition 2 Construction via residue classes

2.1 Residue classes 2.2 Dirichlet characters

3 A few character tables

3.1 Modulus 1 3.2 Modulus 2 3.3 Modulus 3 3.4 Modulus 4 3.5 Modulus 5 3.6 Modulus 6 3.7 Modulus 7 3.8 Modulus 8 3.9 Modulus 9 3.10 Modulus 10 3.11 Modulus 11 3.12 Modulus 12

4 Examples 5 Primitive characters and conductor 6 Character orthogonality 7 History 8 See also 9 References 10 External links

Axiomatic definition[edit] A Dirichlet character is any function

χ

displaystyle chi

from the integers

Z

displaystyle mathbb Z

to the complex numbers

C

displaystyle mathbb C

such that

χ

displaystyle chi

has the following properties:[1]

There exists a positive integer k such that χ(n) = χ(n + k) for all n. If gcd(n,k) > 1 then χ(n) = 0; if gcd(n,k) = 1 then χ(n) ≠ 0. χ(mn) = χ(m)χ(n) for all integers m and n.

From this definition, several other properties can be deduced. By property 3), χ(1)=χ(1×1)=χ(1)χ(1). Since gcd(1, k) = 1, property 2) says χ(1) ≠ 0, so

χ(1) = 1.

Properties 3) and 4) show that every Dirichlet character χ is completely multiplicative. Property 1) says that a character is periodic with period k; we say that

χ

displaystyle chi

is a character to the modulus k. This is equivalent to saying that

If a ≡ b (mod k) then χ(a) = χ(b).

If gcd(a,k) = 1, Euler's theorem says that aφ(k) ≡ 1 (mod k) (where φ(k) is the totient function). Therefore by 5) and 4), χ(aφ(k)) = χ(1) = 1, and by 3), χ(aφ(k)) =χ(a)φ(k). So

For all a relatively prime to k, χ(a) is a φ(k)-th complex root of unity, i.e.

e

2 r i π

/

φ ( k )

displaystyle e^ 2ripi /varphi (k)

for some integer 0 ≤ r < φ(k).

The unique character of period 1 is called the trivial character. Note that any character vanishes at 0 except the trivial one, which is 1 on all integers. A character is called principal if it assumes the value 1 for arguments coprime to its modulus and otherwise is 0.[2] A character is called real if it assumes real values only. A character which is not real is called complex.[3] The sign of the character

χ

displaystyle chi

depends on its value at −1. Specifically,

χ

displaystyle chi

is said to be odd if

χ ( − 1 ) = − 1

displaystyle chi (-1)=-1

and even if

χ ( − 1 ) = 1

displaystyle chi (-1)=1

. Construction via residue classes[edit] Dirichlet characters may be viewed in terms of the character group of the unit group of the ring Z/kZ, as extended residue class characters.[4] Residue classes[edit] Given an integer k, one defines the residue class of an integer n as the set of all integers congruent to n modulo k:

n ^

=

m ∣ m ≡ n

mod

k

.

displaystyle hat n = mmid mequiv nmod k .

That is, the residue class

n ^

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

φ ( k )

displaystyle varphi (k)

, where group multiplication is given by

m n

^

=

m ^

n ^

displaystyle widehat mn = hat m hat n

and

φ

displaystyle varphi

again denotes Euler's phi function. The identity in this group is the residue class

1 ^

displaystyle hat 1

and the inverse of

m ^

displaystyle hat m

is the residue class

n ^

displaystyle hat n

where

m ^

n ^

=

1 ^

displaystyle hat m hat n = hat 1

, i.e.,

m n ≡ 1

mod

k

displaystyle mnequiv 1mod k

. For example, for k=6, the set of units is

1 ^

,

5 ^

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*.[5] Dirichlet characters[edit] 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

χ

displaystyle chi

from (Z/kZ)* to the non-zero complex numbers

χ : (

Z

/

k

Z

)

C

displaystyle chi :(mathbb Z /kmathbb 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

χ

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

χ

0

displaystyle chi _ 0

modulo k has the properties[7]

χ

0

( n ) = 1

displaystyle chi _ 0 (n)=1

if gcd(n, k) = 1 and

χ

0

( n ) = 0

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 few character tables[edit] The tables below help illustrate the nature of a Dirichlet character. They present all of the characters from modulus 1 to modulus 12. The characters χ0 are the principal characters. Modulus 1[edit] There is

φ ( 1 ) = 1

displaystyle varphi (1)=1

character modulo 1:

χ  n     0  

χ

0

( n )

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

χ

0

( n )

displaystyle chi _ 0 (n)

is the Riemann zeta function

ζ ( s ) =

n = 1

1

n

s

=

1

1

s

+

1

2

s

+

1

3

s

+

1

4

s

+ ⋯

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

.

Modulus 2[edit] There is

φ ( 2 ) = 1

displaystyle varphi (2)=1

character modulo 2:

χ  n     0     1  

χ

0

( n )

displaystyle chi _ 0 (n)

0 1

Note that χ is wholly determined by χ(1) since 1 generates the group of units modulo 2. Modulus 3[edit] There are

φ ( 3 ) = 2

displaystyle varphi (3)=2

characters modulo 3:

χ  n     0     1     2  

χ

0

( n )

displaystyle chi _ 0 (n)

0 1 1

χ

1

( n )

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. Modulus 4[edit] There are

φ ( 4 ) = 2

displaystyle varphi (4)=2

characters modulo 4:

χ  n     0     1     2     3  

χ

0

( n )

displaystyle chi _ 0 (n)

0 1 0 1

χ

1

( n )

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

χ

0

( n )

displaystyle chi _ 0 (n)

is the Dirichlet lambda function (closely related to the Dirichlet eta function)

L (

χ

0

, s ) = ( 1 −

2

− s

) ζ ( s )

displaystyle L(chi _ 0 ,s)=(1-2^ -s )zeta (s),

where

ζ ( s )

displaystyle zeta (s)

is the Riemann zeta-function. The L-series for

χ

1

( n )

displaystyle chi _ 1 (n)

is the Dirichlet beta-function

L (

χ

1

, s ) = β ( s ) .

displaystyle L(chi _ 1 ,s)=beta (s).,

Modulus 5[edit] There are

φ ( 5 ) = 4

displaystyle varphi (5)=4

characters modulo 5. In the table below, i is the imaginary unit.

χ  n     0     1     2     3     4  

χ

0

( n )

displaystyle chi _ 0 (n)

0 1 1 1 1

χ

1

( n )

displaystyle chi _ 1 (n)

0 1 i −i −1

χ

2

( n )

displaystyle chi _ 2 (n)

0 1 −1 −1 1

χ

3

( n )

displaystyle chi _ 3 (n)

0 1 −i i −1

Note that χ is wholly determined by χ(2) since 2 generates the group of units modulo 5. Modulus 6[edit] There are

φ ( 6 ) = 2

displaystyle varphi (6)=2

characters modulo 6:

χ  n     0     1     2     3     4     5  

χ

0

( n )

displaystyle chi _ 0 (n)

0 1 0 0 0 1

χ

1

( n )

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. Modulus 7[edit] There are

φ ( 7 ) = 6

displaystyle varphi (7)=6

characters modulo 7. In the table below,

ω =

e

i π

/

3

.

displaystyle omega =e^ ipi /3 .

χ  n     0     1     2     3     4     5     6  

χ

0

( n )

displaystyle chi _ 0 (n)

0 1 1 1 1 1 1

χ

1

( n )

displaystyle chi _ 1 (n)

0 1 ω2 ω −ω −ω2 −1

χ

2

( n )

displaystyle chi _ 2 (n)

0 1 −ω ω2 ω2 −ω 1

χ

3

( n )

displaystyle chi _ 3 (n)

0 1 1 −1 1 −1 −1

χ

4

( n )

displaystyle chi _ 4 (n)

0 1 ω2 −ω −ω ω2 1

χ

5

( n )

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. Modulus 8[edit] There are

φ ( 8 ) = 4

displaystyle varphi (8)=4

characters modulo 8.

χ  n     0     1     2     3     4     5     6     7  

χ

0

( n )

displaystyle chi _ 0 (n)

0 1 0 1 0 1 0 1

χ

1

( n )

displaystyle chi _ 1 (n)

0 1 0 1 0 −1 0 −1

χ

2

( n )

displaystyle chi _ 2 (n)

0 1 0 −1 0 1 0 −1

χ

3

( n )

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. Modulus 9[edit] There are

φ ( 9 ) = 6

displaystyle varphi (9)=6

characters modulo 9. In the table below,

ω =

e

i π

/

3

.

displaystyle omega =e^ ipi /3 .

χ  n     0     1     2     3     4     5     6     7     8  

χ

0

( n )

displaystyle chi _ 0 (n)

0 1 1 0 1 1 0 1 1

χ

1

( n )

displaystyle chi _ 1 (n)

0 1 ω 0 ω2 −ω2 0 −ω −1

χ

2

( n )

displaystyle chi _ 2 (n)

0 1 ω2 0 −ω −ω 0 ω2 1

χ

3

( n )

displaystyle chi _ 3 (n)

0 1 −1 0 1 −1 0 1 −1

χ

4

( n )

displaystyle chi _ 4 (n)

0 1 −ω 0 ω2 ω2 0 −ω 1

χ

5

( n )

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. Modulus 10[edit] There are

φ ( 10 ) = 4

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  

χ

0

( n )

displaystyle chi _ 0 (n)

0 1 0 1 0 0 0 1 0 1

χ

1

( n )

displaystyle chi _ 1 (n)

0 1 0 i 0 0 0 −i 0 −1

χ

2

( n )

displaystyle chi _ 2 (n)

0 1 0 −1 0 0 0 −1 0 1

χ

3

( n )

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. Modulus 11[edit] There are

φ ( 11 ) = 10

displaystyle varphi (11)=10

characters modulo 11. In the table below,

ω =

e

i π

/

5

.

displaystyle omega =e^ ipi /5 .

χ  n     0     1     2     3     4     5     6     7     8     9     10  

χ

0

( n )

displaystyle chi _ 0 (n)

0 1 1 1 1 1 1 1 1 1 1

χ

1

( n )

displaystyle chi _ 1 (n)

0 1 ω −ω3 ω2 ω4 −ω4 −ω2 ω3 −ω −1

χ

2

( n )

displaystyle chi _ 2 (n)

0 1 ω2 −ω ω4 −ω3 −ω3 ω4 −ω ω2 1

χ

3

( n )

displaystyle chi _ 3 (n)

0 1 ω3 ω4 −ω ω2 −ω2 ω −ω4 −ω3 −1

χ

4

( n )

displaystyle chi _ 4 (n)

0 1 ω4 ω2 −ω3 −ω −ω −ω3 ω2 ω4 1

χ

5

( n )

displaystyle chi _ 5 (n)

0 1 −1 1 1 1 −1 −1 −1 1 −1

χ

6

( n )

displaystyle chi _ 6 (n)

0 1 −ω −ω3 ω2 ω4 ω4 ω2 −ω3 −ω 1

χ

7

( n )

displaystyle chi _ 7 (n)

0 1 −ω2 −ω ω4 −ω3 ω3 −ω4 ω ω2 −1

χ

8

( n )

displaystyle chi _ 8 (n)

0 1 −ω3 ω4 −ω ω2 ω2 −ω ω4 −ω3 1

χ

9

( n )

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. Modulus 12[edit] There are

φ ( 12 ) = 4

displaystyle varphi (12)=4

characters modulo 12.

χ  n     0     1     2     3     4     5     6     7     8     9     10     11  

χ

0

( n )

displaystyle chi _ 0 (n)

0 1 0 0 0 1 0 1 0 0 0 1

χ

1

( n )

displaystyle chi _ 1 (n)

0 1 0 0 0 1 0 −1 0 0 0 −1

χ

2

( n )

displaystyle chi _ 2 (n)

0 1 0 0 0 −1 0 1 0 0 0 −1

χ

3

( n )

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. Examples[edit] If p is an odd prime number, then the function

χ ( n ) =

(

n p

)

,  

displaystyle chi (n)=left( frac n p right),

where

(

n p

)

displaystyle left( frac n p right)

is the Legendre symbol, is a primitive Dirichlet character modulo p.[9]

More generally, if m is a positive odd number, the function

χ ( n ) =

(

n m

)

,  

displaystyle chi (n)=left( frac n m right),

where

(

n m

)

displaystyle left( frac n m right)

is the Jacobi symbol, is a Dirichlet character modulo m.[9]

These are examples of real characters. In general, all real characters arise from the Kronecker symbol. Primitive characters and conductor[edit] Residues mod N give rise to residues mod M, for any factor M of N, by discarding some information. The effect on Dirichlet characters goes in the opposite direction: if χ is a character mod M, it induces a character χ* mod N for any multiple N of M. A character is primitive if it is not induced by any character of smaller modulus.[3] If χ is a character mod n and d divides n, then we say that the modulus d is an induced modulus for χ if a coprime to n and 1 mod d implies χ(a)=1:[10] equivalently, χ(a) = χ(b) whenever a, b are congruent mod d and each coprime to n.[11] A character is primitive if there is no smaller induced modulus.[11] We can formalize this differently by defining characters χ1 mod N1 and χ2 mod N2 to be co-trained if for some modulus N such that N1 and N2 both divide N we have χ1(n) = χ2(n) for all n coprime to N: that is, there is some character χ* induced by each of χ1 and χ2. This is an equivalence relation on characters. A character with the smallest modulus in an equivalence class is primitive and this smallest modulus is the conductor of the characters in the class. Imprimitivity of characters can lead to missing Euler factors in their L-functions. Character orthogonality[edit] The orthogonality relations for characters of a finite group transfer to Dirichlet characters.[12] If we fix a character χ modulo n then the sum

a

mod

n

χ ( a ) = 0  

displaystyle sum _ a bmod n chi (a)=0

unless χ is principal, in which case the sum is φ(n). Similarly, if we fix a residue class a modulo n and sum over all characters we have

χ

χ ( a ) = 0  

displaystyle sum _ chi chi (a)=0

unless

a ≡ 1

( mod

n )

displaystyle aequiv 1 pmod n

in which case the sum is φ(n). We deduce that any periodic function with period n supported on the residue classes prime to n is a linear combination of Dirichlet characters.[13] History[edit] Dirichlet characters and their L-series were introduced by Peter Gustav Lejeune Dirichlet, in 1831, in order to prove Dirichlet's theorem on arithmetic progressions. He only studied them for real s and especially as s tends to 1. The extension of these functions to complex s in the whole complex plane was obtained by Bernhard Riemann in 1859. See also[edit]

Hecke character (also known as grössencharacter) Character sum Gaussian sum Multiplicative group of integers modulo n Primitive root modulo n Selberg class

References[edit]

^ Montgomery & Vaughan (2007) pp.117–8 ^ Montgomery & Vaughan (2007) p.115 ^ a b Montgomery & Vaughan (2007) p.123 ^ Fröhlich & Taylor (1991) p.218 ^ Frohlich & Taylor (1991) p.215 ^ Apostol (1976) p.139 ^ a b c Apostol (1976) p.138 ^ Apostol (1976) p.134 ^ a b Montgomery & Vaughan (2007) p.295 ^ Apostol (1976) p.166 ^ a b Apostol (1976) p.168 ^ Apostol (1976) p.140 ^ Davenport (1967) pp.31–32

See chapter 6 of Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001  Apostol, T. M. (1971). "Some properties of completely multiplicative arithmetical functions". The American Mathematical Monthly. 78 (3): 266–271. doi:10.2307/2317522. JSTOR 2317522. MR 0279053. Zbl 0209.34302.  Davenport, Harold (1967). Multiplicative number theory. Lectures in advanced mathematics. 1. Chicago: Markham. Zbl 0159.06303.  Hasse, Helmut (1964). Vorlesungen über Zahlentheorie. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen. 59 (2nd revised ed.). Springer-Verlag. MR 0188128. Zbl 0123.04201.  see chapter 13. Mathar, R. J. (2010). "Table of Dirichlet L-series and prime zeta modulo functions for small moduli". arXiv:1008.2547  [math.NT].  Montgomery, Hugh L; Vaughan, Robert C. (2007). Multiplicative number theory. I. Classical theory. Cambridge Studies in Advanced Mathematics. 97. Cambridge University Press. ISBN 0-521-84903-9. Zbl 1142.11001.  Spira, Robert (1969). "Calculation of Dirichlet L-Functions". Mathematics of Computation. 23 (107): 489–497. doi:10.1090/S0025-5718-1969-0247742-X. MR 0247742. Zbl 0182.07001.  Fröhlich, A.; Taylor, M.J. (1991). Algebraic number theory. Cambridge studies in advanced mathematics. 27. Cambridge University Press. ISBN 0-521-36664-X. Zbl 0744.11001. 

External links[edit]

Hazewinkel, Michiel, ed. (2001) [1994], "Dirichlet character", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4  "Dirichlet Characters". 

.