Dirichlet character
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In number theory, Dirichlet characters are certain arithmetic functions which arise from completely multiplicative characters on the units of . Dirichlet characters are used to define Dirichlet Lfunctions, which are meromorphic functions with a variety of interesting analytic properties. If is a Dirichlet character, one defines its Dirichlet Lseries by
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 Lfunctions are generalizations of the Riemann zetafunction and appear prominently in the generalized Riemann hypothesis.
Dirichlet characters are named in honour of Peter Gustav Lejeune Dirichlet.
Contents
Axiomatic definition
A Dirichlet character is any function from the integers to the complex numbers such that 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 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 a^{φ(k)} ≡ 1 (mod k) (where φ(k) is the totient function). Therefore, by 5) and 4), χ(a^{φ(k)}) = χ(1) = 1, and by 3), χ(a^{φ(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. 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 depends on its value at −1. Specifically, is said to be odd if and even if .
Construction via residue classes
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
Given an integer k, one defines the residue class of an integer n as the set of all integers congruent to n modulo k: That is, the residue class is the coset of n in the quotient ring Z/kZ.
The set of units modulo k forms an abelian group of order , where group multiplication is given by and again denotes Euler's phi function. The identity in this group is the residue class and the inverse of is the residue class where , i.e., . For example, for k=6, the set of units is 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
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 from (Z/kZ)^{*} to the nonzero complex numbers
 ,
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 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 nontrivial factor in common with k. The resulting function will then be a Dirichlet character.^{[7]}
The principal character modulo k has the properties^{[7]}
 if gcd(n, k) = 1 and
 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 nontrivial common factor with k and is 1 at other integers.
There are φ(n) Dirichlet characters modulo n.^{[7]}
A few character tables
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 χ_{0} are the principal characters.
Modulus 1
There is character modulo 1:
χ \ n 0 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 Lseries for is the Riemann zeta function
 .
Modulus 2
There is character modulo 2:
χ \ n 0 1 0 1
Note that χ is wholly determined by χ(1) since 1 generates the group of units modulo 2.
The Dirichlet Lseries for is the Dirichlet lambda function (closely related to the Dirichlet eta function)
Modulus 3
There are characters modulo 3:
χ \ n 0 1 2 0 1 1 0 1 −1
Note that χ is wholly determined by χ(2) since 2 generates the group of units modulo 3.
Modulus 4
There are characters modulo 4:
χ \ n 0 1 2 3 0 1 0 1 0 1 0 −1
Note that χ is wholly determined by χ(3) since 3 generates the group of units modulo 4.
The Dirichlet Lseries for is the Dirichlet lambda function (closely related to the Dirichlet eta function)
where is the Riemann zetafunction. The Lseries for is the Dirichlet betafunction
Modulus 5
There are characters modulo 5. In the table below, i is the imaginary unit.
χ \ n 0 1 2 3 4 0 1 1 1 1 0 1 i −i −1 0 1 −1 −1 1 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
There are characters modulo 6:
χ \ n 0 1 2 3 4 5 0 1 0 0 0 1 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
There are characters modulo 7. In the table below,
χ \ n 0 1 2 3 4 5 6 0 1 1 1 1 1 1 0 1 ω^{2} ω −ω −ω^{2} −1 0 1 −ω ω^{2} ω^{2} −ω 1 0 1 1 −1 1 −1 −1 0 1 ω^{2} −ω −ω ω^{2} 1 0 1 −ω −ω^{2} ω^{2} ω −1
Note that χ is wholly determined by χ(3) since 3 generates the group of units modulo 7.
Modulus 8
There are characters modulo 8.
χ \ n 0 1 2 3 4 5 6 7 0 1 0 1 0 1 0 1 0 1 0 1 0 −1 0 −1 0 1 0 −1 0 1 0 −1 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
There are characters modulo 9. In the table below,
χ \ n 0 1 2 3 4 5 6 7 8 0 1 1 0 1 1 0 1 1 0 1 ω 0 ω^{2} −ω^{2} 0 −ω −1 0 1 ω^{2} 0 −ω −ω 0 ω^{2} 1 0 1 −1 0 1 −1 0 1 −1 0 1 −ω 0 ω^{2} ω^{2} 0 −ω 1 0 1 −ω^{2} 0 −ω ω 0 ω^{2} −1
Note that χ is wholly determined by χ(2) since 2 generates the group of units modulo 9.
Modulus 10
There are characters modulo 10. In the table below, i is the imaginary unit.
χ \ n 0 1 2 3 4 5 6 7 8 9 0 1 0 1 0 0 0 1 0 1 0 1 0 i 0 0 0 −i 0 −1 0 1 0 −1 0 0 0 −1 0 1 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
There are characters modulo 11. In the table below,
χ \ n 0 1 2 3 4 5 6 7 8 9 10 0 1 1 1 1 1 1 1 1 1 1 0 1 ω −ω^{3} ω^{2} ω^{4} −ω^{4} −ω^{2} ω^{3} −ω −1 0 1 ω^{2} −ω ω^{4} −ω^{3} −ω^{3} ω^{4} −ω ω^{2} 1 0 1 ω^{3} ω^{4} −ω ω^{2} −ω^{2} ω −ω^{4} −ω^{3} −1 0 1 ω^{4} ω^{2} −ω^{3} −ω −ω −ω^{3} ω^{2} ω^{4} 1 0 1 −1 1 1 1 −1 −1 −1 1 −1 0 1 −ω −ω^{3} ω^{2} ω^{4} ω^{4} ω^{2} −ω^{3} −ω 1 0 1 −ω^{2} −ω ω^{4} −ω^{3} ω^{3} −ω^{4} ω ω^{2} −1 0 1 −ω^{3} ω^{4} −ω ω^{2} ω^{2} −ω ω^{4} −ω^{3} 1 0 1 −ω^{4} ω^{2} −ω^{3} −ω ω ω^{3} −ω^{2} ω^{4} −1
Note that χ is wholly determined by χ(2) since 2 generates the group of units modulo 11.
Modulus 12
There are characters modulo 12.
χ \ n 0 1 2 3 4 5 6 7 8 9 10 11 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 −1 0 0 0 −1 0 1 0 0 0 −1 0 1 0 0 0 −1 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
If p is an odd prime number, then the function
 where is the Legendre symbol, is a primitive Dirichlet character modulo p.^{[9]}
More generally, if m is a positive odd number, the function
 where 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
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 χ_{1} mod N_{1} and χ_{2} mod N_{2} to be cotrained if for some modulus N such that N_{1} and N_{2} both divide N we have χ_{1}(n) = χ_{2}(n) for all n coprime to N: that is, there is some character χ* induced by each of χ_{1} and χ_{2}. 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 Lfunctions.
Character orthogonality
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
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
unless 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]} We also have the a character sum relation given in Chapter 4 of Davenport given by
where the sum is taken over all Dirichlet characters modulo some fixed q, a and n are fixed with , and denotes Euler's totient function.
History
Dirichlet characters and their Lseries 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
 Hecke character (also known as grössencharacter)
 Character sum
 Gaussian sum
 Multiplicative group of integers modulo n
 Primitive root modulo n
 Selberg class
References
 ^ 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 YorkHeidelberg: SpringerVerlag, ISBN 9780387901633, 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.). SpringerVerlag. MR 0188128. Zbl 0123.04201. see chapter 13.
 Mathar, R. J. (2010). "Table of Dirichlet Lseries 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 0521849039. Zbl 1142.11001.
 Spira, Robert (1969). "Calculation of Dirichlet LFunctions". Mathematics of Computation. 23 (107): 489–497. doi:10.1090/S0025571819690247742X. MR 0247742. Zbl 0182.07001.
 Fröhlich, A.; Taylor, M.J. (1991). Algebraic number theory. Cambridge studies in advanced mathematics. 27. Cambridge University Press. ISBN 052136664X. Zbl 0744.11001.
External links
 Hazewinkel, Michiel, ed. (2001) [1994], "Dirichlet character", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 9781556080104
 "Dirichlet Characters". in the LMFDB