- Dirichlet character
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 L-functions, which are meromorphic functions with a variety of interesting analytic properties. If χ is a Dirichlet character, one defines its Dirichlet L-series 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 L-functions are generalizations of the Riemann zeta-function and appear prominently in the generalized Riemann hypothesis.
Dirichlet characters are named in honour of Johann Peter Gustav Lejeune Dirichlet.
- 1 Axiomatic definition
- 2 Construction via residue classes
- 3 A few character tables
- 4 Examples
- 5 Conductors
- 6 History
- 7 See also
- 8 References
- 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 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.
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. A character is called real if it assumes real values only. A character which is not real is called complex.
The sign of the character χ depends on its value at −1. Specifically, χ is said to be odd if χ(−1) = −1 and even if χ(−1) = 1.
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 given below.
The set of units modulo k forms an abelian group of order φ(k), where group multiplication is given by and ϕ 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.
A Dirichlet character modulo k is a group homomorphism χ from the unit group modulo k to the non-zero complex numbers
necessarily with values that are roots of unity since the units modulo k form a finite group. We can lift χ to a completely multiplicative function on integers relatively prime to k and then to all integers by extending the function to be 0 on integers having a non-trivial factor in common with k. The principal character χ1 modulo k has the properties
- χ1(n) = 1 if gcd(n, k) = 1 and
- χ1(n) = 0 if gcd(n, k) > 1.
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.
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 10. The characters χ1 are the principal characters.
There is ϕ(1) = 1 character modulo 1:
χ \ n 0 χ1(n) 1
This is the trivial character.
There is ϕ(2) = 1 character modulo 2:
χ \ n 0 1 χ1(n) 0 1
Note that χ is wholly determined by χ(1) since 1 generates the group of units modulo 2.
There are ϕ(3) = 2 characters modulo 3:
χ \ n 0 1 2 χ1(n) 0 1 1 χ2(n) 0 1 −1
Note that χ is wholly determined by χ(2) since 2 generates the group of units modulo 3.
There are ϕ(4) = 2 characters modulo 4:
χ \ n 0 1 2 3 χ1(n) 0 1 0 1 χ2(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 χ1(n) is the Dirichlet lambda function (closely related to the Dirichlet eta function)
where ζ(s) is the Riemann zeta-function. The L-series for χ2(n) is the Dirichlet beta-function
There are ϕ(5) = 4 characters modulo 5. In the tables, i is a square root of − 1.
χ \ n 0 1 2 3 4 χ1(n) 0 1 1 1 1 χ2(n) 0 1 i −i −1 χ3(n) 0 1 −1 −1 1 χ4(n) 0 1 −i i −1
Note that χ is wholly determined by χ(2) since 2 generates the group of units modulo 5.
There are ϕ(6) = 2 characters modulo 6:
χ \ n 0 1 2 3 4 5 χ1(n) 0 1 0 0 0 1 χ2(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.
There are ϕ(7) = 6 characters modulo 7. In the table below, ω = exp(πi / 3).
χ \ n 0 1 2 3 4 5 6 χ1(n) 0 1 1 1 1 1 1 χ2(n) 0 1 ω2 ω −ω −ω2 −1 χ3(n) 0 1 −ω ω2 ω2 −ω 1 χ4(n) 0 1 1 −1 1 −1 −1 χ5(n) 0 1 ω2 −ω −ω ω2 1 χ6(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.
There are ϕ(8) = 4 characters modulo 8.
χ \ n 0 1 2 3 4 5 6 7 χ1(n) 0 1 0 1 0 1 0 1 χ2(n) 0 1 0 1 0 −1 0 −1 χ3(n) 0 1 0 −1 0 1 0 −1 χ4(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.
There are ϕ(9) = 6 characters modulo 9. In the table below, ω = exp(πi / 3).
χ \ n 0 1 2 3 4 5 6 7 8 χ1(n) 0 1 1 0 1 1 0 1 1 χ2(n) 0 1 ω 0 ω2 −ω2 0 −ω −1 χ3(n) 0 1 ω2 0 −ω −ω 0 ω2 1 χ4(n) 0 1 −1 0 1 −1 0 1 −1 χ5(n) 0 1 −ω 0 ω2 ω2 0 −ω 1 χ6(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.
There are ϕ(10) = 4 characters modulo 10.
χ \ n 0 1 2 3 4 5 6 7 8 9 χ1(n) 0 1 0 1 0 0 0 1 0 1 χ2(n) 0 1 0 i 0 0 0 −i 0 −1 χ3(n) 0 1 0 −1 0 0 0 −1 0 1 χ4(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.
If p is a prime number, then the function
- where is the Legendre symbol, is a Dirichlet character modulo p.
More generally, if m is an odd number the function
- where is the Jacobi symbol, is a Dirichlet character modulo m. These are called the quadratic characters.
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 gives rise to a character χ* mod N for any multiple N of M. With some attention to the values at which characters take the value 0, one gets the concept of a primitive Dirichlet character, one that does not arise from a factor; and the associated idea of conductor, i.e. the natural (smallest) modulus for a character. Imprimitive characters can cause missing Euler factors in L-functions.
Dirichlet characters and their L-series were introduced by Johann 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 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, MR0434929
- Spira, Robert (1969). "Calculation of Dirichlet L-Functions". Mathematics of Computation 23 (107): 489–497. doi:10.1090/S0025-5718-1969-0247742-X. MR0247742.
- 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. MR0279053.
- Hasse, Helmut (1964). Vorlesungen über Zahlentheorie. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen. Springer. MR0188128. see chapter 13.
Wikimedia Foundation. 2010.
Look at other dictionaries:
Dirichlet L-function — In mathematics, a Dirichlet L series is a function of the form Here χ is a Dirichlet character and s a complex variable with real part greater than 1. By analytic continuation, this function can be extended to a meromorphic function on the whole… … Wikipedia
Dirichlet series — In mathematics, a Dirichlet series is any series of the form where s and an are complex numbers and n = 1, 2, 3, ... . It is a special case of general Dirichlet series. Dirichlet series play a variety of important roles in analytic number theory … Wikipedia
Character (mathematics) — In mathematics, a character is (most commonly) a special kind of function from a group to a field (such as the complex numbers). There are at least two distinct, but overlapping meanings. Other uses of the word character are almost always… … Wikipedia
Character sum — In mathematics, a character sum is a sum of values of a Dirichlet character χ modulo N, taken over a given range of values of n. Such sums are basic in a number of questions, for example in the distribution of quadratic residues, and in… … Wikipedia
Dirichlet beta function — This article is about the Dirichlet beta function. For other beta functions, see Beta function (disambiguation). In mathematics, the Dirichlet beta function (also known as the Catalan beta function) is a special function, closely related to the… … Wikipedia
Character group — In mathematics, a character group is the group of representations of a group by complex valued functions. These functions can be thought of as one dimensional matrix representations and so are special cases of the group characters which arises in … Wikipedia
Hecke character — In mathematics, in the field of number theory, a Hecke character is a generalisation of a Dirichlet character, introduced by Erich Hecke to construct a class of L functions larger than Dirichlet L functions, and a natural setting for the Dedekind … Wikipedia
Función beta de Dirichlet — Este artículo trata sobre función beta de Dirichlet. Para otras funciones beta, véase Función beta (desambiguación). En matemática, la función beta de Dirichlet (también conocida como la función beta de Catalan) es una función especial,… … Wikipedia Español
Generalized Riemann hypothesis — The Riemann hypothesis is one of the most important conjectures in mathematics. It is a statement about the zeros of the Riemann zeta function. Various geometrical and arithmetical objects can be described by so called global L functions, which… … Wikipedia
Quadratic residue — In number theory, an integer q is called a quadratic residue modulo n if it is congruent to a perfect square modulo n; i.e., if there exists an integer x such that: Otherwise, q is called a quadratic nonresidue modulo n. Originally an abstract… … Wikipedia