Dirichlet character

From Wikipedia, the free encyclopedia
Jump to navigation Jump to search

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)     i.e. is completely multiplicative.
2)  
3)   ; i.e. is periodic with period .

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

Dirichlet introduced these functions in his 1837 paper on primes in arithmetic progressions.[3][4]

Notation[edit]

is the Euler 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 are 1) an isomorphism and 2) 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 deternined 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 [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 [edit]

The only character mod 2 is the principal character .

−1 is 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 be the factorization of into powers of primes. It is known that[16]

For define[17]

Then for and all and

showing that is a character and
showing an isomorphism

Examples [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 [18]

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,[19][20]

Orthogonality[edit]

The two orthogonality relations are

    and    

where the first sum has one summand per residue class.[21]

The relations can be written in the symmetric form

    and    

The identity for shows that the relations are equivalent to each other.

The first relation is easy to prove: If there are non-zero summands each equal to 1. Otherwise[22] there is some  Then

[23]   implying

The first factor is not zero, therefore the second one is. Since the relations are equivalent, the second one is also proved. QED

The second relation can be proven directly in the same way, but requires a lemma[24]

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.[25][26]

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[27]

has period 16, but has period 8 and has period 4:   and   The smallest prime power for which is periodic is the 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 5 and those of have period 3. 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.[28]

The character is primitive if and only if each of the factors is primitive.[29]

Primitive characters are often necessary (or at least simplify) formulas in the theories of L-functions[30] 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.[31]

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.[32][33]

Real characters are Kronecker symbols;[34] for example, the principal character can be written[35] .

The real characters in the examples are:

Principal[edit]

If the principal character is[36]

             

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[29] they are imaginary.[37]

                   

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[38]

Let and let be primitive.

If

[39]

define

,[40]  

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[41]

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 [42]

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.[43]

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[44] in 1956, and proved in 2017 by Klurman and Mangerel.[45]

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. ^ See Conrey labeling for details
  18. ^ This is true for all finite abelian groups: ; See Ireland & Rosen pp. 253-254
  19. ^ because the formulas for mod prime powers are symmetric in and and the formula for products preserves this symmetry. See Davenport, p. 29.
  20. ^ 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.
  21. ^ See #Relation to group characters above.
  22. ^ by the definition of
  23. ^ because multiplying every element in a group by a constant element merely permutes the elements. See Group (mathematics)
  24. ^ Davenport p. 30 (paraphrase) To prove [the second relation] one has to use ideas that we have used in the construction (as in Landau pp. 109-114), or appeal to the basis theorem for abelian groups (as in Ireland & Rosen pp. 253-254)
  25. ^ Davenport chs. 1, 4; Landau p. 114
  26. ^ Note that if is any function ; see Fourier transform on finite groups#Fourier transform for finite abelian groups
  27. ^ This section follows Davenport pp. 35-36,
  28. ^ Davenport classifies it as neither primitive nor imprimitive; the LMFDB induces it from
  29. ^ a b Note that if is two times an odd number, , all characters mod are imprimitive because
  30. ^ For example the functional equation of is only valid for primitive . See Davenport, p. 85
  31. ^ In fact, for prime modulus is the Legendre symbol: Sketch of proof: is even (odd) if a is a quadratic residue (nonresidue)
  32. ^ Davenport, chs. 1, 4.
  33. ^ Ireland and Rosen's proof, valid for all moduli, also has these two cases. pp. 259 ff
  34. ^ Davenport p. 40
  35. ^ The notation is a shorter way of writing
  36. ^ The product of primes ensures it is zero if ; the squares ensure its only nonzero value is 1.
  37. ^ Davenport pp. 38-40
  38. ^ Koblittz, prop. 17b p. 127
  39. ^ means 1) where and and 2) where and See Koblitz Ch. III.
  40. ^ the twist of by
  41. ^ LMFDB definition of Kloosterman sum
  42. ^ Davenport p. 30
  43. ^ Sarkozy
  44. ^ Chudakov
  45. ^ 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. 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. 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.

External links[edit]

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