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]

  1. that is, is completely multiplicative.
  2. (gcd is the greatest common divisor)
  3. ; that is, is periodic with period .

The simplest possible character, called the principal character, usually denoted , (see Notation below) exists for all moduli:[2]

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]

Notation

edit

  is Euler's totient function.

  is a complex primitive n-th root of unity:

  but  

  is the group of units mod  . It has order  

  is the group of Dirichlet characters mod  .

  etc. are prime numbers.

  is a standard[5] abbreviation[6] for  

  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

edit

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]

If   is a finite abelian group then[8] there is an isomorphism  , and the orthogonality relations:[9]

      and      

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

edit

4) Since   property 2) says   so it can be canceled from both sides of  :

 [11]

5) Property 3) is equivalent to

if     then  

6) Property 1) implies that, for any positive integer  

 

7) Euler's theorem states that if   then   Therefore,

 

That is, the nonzero values of   are  -th roots of unity:

 

for some integer   which depends on   and  . This implies there are only a finite number of characters for a given modulus.

8) If   and   are two characters for the same modulus so is their product   defined by pointwise multiplication:

    (  obviously satisfies 1-3).[12]

The principal character is an identity:

 

9) Let   denote the inverse of   in  . Then

  so   which extends 6) to all integers.

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

edit

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

edit

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

edit

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

edit

  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

edit

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

edit

Let   where   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  

The Chinese remainder theorem (CRT) implies that the   are simply  

There are subgroups   such that [17]

  and
 

Then   and every   corresponds to a  -tuple   where   and   Every   can be uniquely factored as   [18] [19]

If   is a character mod   on the subgroup   it must be identical to some   mod   Then

 

showing that every character mod   is the product of characters mod the  .

For   define[20]

 

Then for   and all   and  [21]

  showing that   is a character and
  showing an isomorphism  


Examples m = 15, 24, 40

edit

 

The factorization of the characters mod 15 is

 

The nonzero values of the characters mod 15 are

 .

  The factorization of the characters mod 24 is

 

The nonzero values of the characters mod 24 are

 .

  The factorization of the characters mod 40 is

 

The nonzero values of the characters mod 40 are

 .

Summary

edit

Let  ,   be the factorization of   and assume  

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  :

 

If   the product   is a character   where   is given by   and  

Also,[23][24]  

Orthogonality

edit

The two orthogonality relations are[25]

      and      

The relations can be written in the symmetric form

      and      

The first relation is easy to prove: If   there are   non-zero summands each equal to 1. If  there is[26] some    Then

 [27]   implying
    Dividing by the first factor gives   QED. The identity   for   shows that the relations are equivalent to each other.

The second relation can be proven directly in the same way, but requires a lemma[28]

Given   there is a  

The second relation has an important corollary: if   define the function

    Then
 

That is   the indicator function of the residue class  . It is basic in the proof of Dirichlet's theorem.[29][30]

Classification of characters

edit

Conductor; Primitive and induced characters

edit

Any character mod a prime power is also a character mod every larger power. For example, mod 16[31]

 

  has period 16, but   has period 8 and   has period 4:     and   

We say that a character   of modulus   has a quasiperiod of   if   for all  ,   coprime to   satisfying   mod  .[32] For example,  , the only Dirichlet character of modulus  , has a quasiperiod of  , but not a period of   (it has a period of  , though). The smallest positive integer for which   is quasiperiodic is the conductor of  .[33] So, for instance,   has a conductor of  .

The conductor of   is 16, the conductor of   is 8 and that of   and   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:   is induced from   and   and   are induced from  .

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,

 .

The nonzero values of   have period 15, but those of   have period 3 and those of   have period 5. This is easier to see by juxtaposing them with characters mod 3 and 5:

 .

If a character mod   is defined as

 ,   or equivalently as  

its nonzero values are determined by the character mod   and have period  .

The smallest period of the nonzero values is the conductor of the character. For example, the conductor of   is 15, the conductor of   is 3, and that of   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,   is induced from   and   is induced from  

The principal character is not primitive.[34]

The character   is primitive if and only if each of the factors is primitive.[35]

Primitive characters often simplify (or make possible) formulas in the theories of L-functions[36] and modular forms.

Parity

edit

  is even if   and is odd if  

This distinction appears in the functional equation of the Dirichlet L-function.

Order

edit

The order of a character is its order as an element of the group  , i.e. the smallest positive integer   such that   Because of the isomorphism   the order of   is the same as the order of   in   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   which is  

Real characters

edit

  is real or quadratic if all of its values are real (they must be  ); otherwise it is complex or imaginary.

  is real if and only if  ;   is real if and only if  ; in particular,   is real and non-principal.[37]

Dirichlet's original proof that   (which was only valid for prime moduli) took two different forms depending on whether   was real or not. His later proof, valid for all moduli, was based on his class number formula.[38][39]

Real characters are Kronecker symbols;[40] for example, the principal character can be written[41]  .

The real characters in the examples are:

Principal

edit

If   the principal character is[42]  

                           

Primitive

edit

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[35] they are imaginary.[43]

                                         

Imprimitive

edit

                           

                   

                   

Applications

edit

L-functions

edit

The Dirichlet L-series for a character   is

 

This series only converges for  ; it can be analytically continued to a meromorphic function

Dirichlet introduced the  -function along with the characters in his 1837 paper.

Modular forms and functions

edit

Dirichlet characters appear several places in the theory of modular forms and functions. A typical example is[44]

Let   and let   be primitive.

If

 [45]

define

 ,[46]  

Then

 . If   is a cusp form so is  

See theta series of a Dirichlet character for another example.

Gauss sum

edit

The Gauss sum of a Dirichlet character modulo N is

 

It appears in the functional equation of the Dirichlet L-function.

Jacobi sum

edit

If   and   are Dirichlet characters mod a prime   their Jacobi sum is

 

Jacobi sums can be factored into products of Gauss sums.

Kloosterman sum

edit

If   is a Dirichlet character mod   and   the Kloosterman sum   is defined as[47]

 

If   it is a Gauss sum.

Sufficient conditions

edit

It is not necessary to establish the defining properties 1) – 3) to show that a function is a Dirichlet character.

From Davenport's book

edit

If   such that

1)    
2)    ,
3)   If   then  , but
4)     is not always 0,

then   is one of the   characters mod  [48]

Sárközy's Condition

edit

A Dirichlet character is a completely multiplicative function   that satisfies a linear recurrence relation: that is, if  

for all positive integer  , where   are not all zero and   are distinct then   is a Dirichlet character.[49]

Chudakov's Condition

edit

A Dirichlet character is a completely multiplicative function   satisfying the following three properties: a)   takes only finitely many values; b)   vanishes at only finitely many primes; c) there is an   for which the remainder

 

is uniformly bounded, as  . This equivalent definition of Dirichlet characters was conjectured by Chudakov[50] in 1956, and proved in 2017 by Klurman and Mangerel.[51]

See also

edit

Notes

edit
  1. ^ This is the standard definition; e.g. Davenport p.27; Landau p. 109; Ireland and Rosen p. 253
  2. ^ Note the special case of modulus 1: the unique character mod 1 is the constant 1; all other characters are 0 at 0
  3. ^ Davenport p. 1
  4. ^ An English translation is in External Links
  5. ^ Used in Davenport, Landau, Ireland and Rosen
  6. ^   is equivalent to  
  7. ^ See Multiplicative character
  8. ^ Ireland and Rosen p. 253-254
  9. ^ See Character group#Orthogonality of characters
  10. ^ Davenport p. 27
  11. ^ These properties are derived in all introductions to the subject, e.g. Davenport p. 27, Landau p. 109.
  12. ^ In general, the product of a character mod   and a character mod   is a character mod  
  13. ^ Except for the use of the modified Conrie labeling, this section follows Davenport pp. 1-3, 27-30
  14. ^ There is a primitive root mod   which is a primitive root mod   and all higher powers of  . See, e.g., Landau p. 106
  15. ^ Landau pp. 107-108
  16. ^ See group of units for details
  17. ^ To construct the   for each   use the CRT to find   where
     
  18. ^ Assume   corresponds to  . By construction   corresponds to  ,   to   etc. whose coordinate-wise product is  
  19. ^ For example let   Then   and   The factorization of the elements of   is
     
  20. ^ See Conrey labeling.
  21. ^ Because these formulas are true for each factor.
  22. ^ This is true for all finite abelian groups:  ; See Ireland & Rosen pp. 253-254
  23. ^ because the formulas for   mod prime powers are symmetric in   and   and the formula for products preserves this symmetry. See Davenport, p. 29.
  24. ^ This is the same thing as saying that the n-th column and the n-th row in the tables of nonzero values are the same.
  25. ^ See #Relation to group characters above.
  26. ^ by the definition of  
  27. ^ because multiplying every element in a group by a constant element merely permutes the elements. See Group (mathematics)
  28. ^ Davenport p. 30 (paraphrase) To prove [the second relation] one has to use ideas that we have used in the construction [as in this article or Landau pp. 109-114], or appeal to the basis theorem for abelian groups [as in Ireland & Rosen pp. 253-254]
  29. ^ Davenport chs. 1, 4; Landau p. 114
  30. ^ Note that if   is any function  ; see Fourier transform on finite groups#Fourier transform for finite abelian groups
  31. ^ This section follows Davenport pp. 35-36,
  32. ^ Platt, Dave. "Dirichlet characters Def. 11.10" (PDF). Retrieved April 5, 2024.
  33. ^ "Conductor of a Dirichlet character (reviewed)". LMFDB. Retrieved April 5, 2024.
  34. ^ Davenport classifies it as neither primitive nor imprimitive; the LMFDB induces it from  
  35. ^ a b Note that if   is two times an odd number,  , all characters mod   are imprimitive because  
  36. ^ For example the functional equation of   is only valid for primitive  . See Davenport, p. 85
  37. ^ In fact, for prime modulus   is the Legendre symbol:   Sketch of proof:   is even (odd) if a is a quadratic residue (nonresidue)
  38. ^ Davenport, chs. 1, 4.
  39. ^ Ireland and Rosen's proof, valid for all moduli, also has these two cases. pp. 259 ff
  40. ^ Davenport p. 40
  41. ^ The notation   is a shorter way of writing  
  42. ^ The product of primes ensures it is zero if  ; the squares ensure its only nonzero value is 1.
  43. ^ Davenport pp. 38-40
  44. ^ Koblittz, prop. 17b p. 127
  45. ^   means 1)   where   and   and 2)   where   and   See Koblitz Ch. III.
  46. ^ the twist of   by  
  47. ^ LMFDB definition of Kloosterman sum
  48. ^ Davenport p. 30
  49. ^ Sarkozy
  50. ^ Chudakov
  51. ^ Klurman

References

edit
  • Chudakov, N.G. "Theory of the characters of number semigroups". J. Indian Math. Soc. 20: 11–15.
  • Davenport, Harold (1967). Multiplicative number theory. Lectures in advanced mathematics. Vol. 1. Chicago: Markham. Zbl 0159.06303.
  • Ireland, Kenneth; Rosen, Michael (1990), A Classical Introduction to Modern Number Theory (Second edition), New York: Springer, ISBN 0-387-97329-X
  • Klurman, Oleksiy; Mangerel, Alexander P. (2017). "Rigidity Theorems for Multiplicative Functions". Math. Ann. 372 (1): 651–697. arXiv:1707.07817. Bibcode:2017arXiv170707817K. doi:10.1007/s00208-018-1724-6. S2CID 119597384.
  • Koblitz, Neal (1993). Introduction to Elliptic Curves and Modular Forms. Graduate Texts in Mathematics. Vol. 97 (2nd revised ed.). Springer-Verlag. ISBN 0-387-97966-2.
  • Landau, Edmund (1966), Elementary Number Theory, New York: Chelsea
  • Sarkozy, Andras. "On multiplicative arithmetic functions satisfying a linear recursion". Studia Sci. Math. Hung. 13 (1–2): 79–104.
edit