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) There exists a positive integer k such that χ(n) = χ(n + k) for all n.
2) If gcd(n,k) > 1 then χ(n) = 0; if gcd(n,k) = 1 then χ(n) ≠ 0.
3) χ(mn) = χ(m)χ(n) for all integers m and n.
These consequences are important:
By property 3), χ(1)=χ(1×1)=χ(1)χ(1); since gcd(1, k) = 1, property 2) says χ(1) ≠ 0, so
4) χ(1) = 1.1) 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
5) 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,
6) χ(a) is a φ(k)-th complex root of unity.
Condition 3) says that a character is completely multiplicative.
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 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.
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.
This is the trivial character.
Note that χ is wholly determined by χ(1) since 1 generates the group of units modulo 2.
Note that χ is wholly determined by χ(2) since 2 generates the group of units modulo 3.
Note that χ is wholly determined by χ(3) since 3 generates the group of units modulo 4.
The Dirichlet L-series for is
where is the Riemann zeta-function. The L-series for is the Dirichlet beta-function
Note that χ is wholly determined by χ(2) since 2 generates the group of units modulo 5.
Note that χ is wholly determined by χ(5) since 5 generates the group of units modulo 6.
Note that χ is wholly determined by χ(3) since 3 generates the group of units modulo 7.
Note that χ is wholly determined by χ(3) and χ(5) since 3 and 5 generate the group of units modulo 8.
Note that χ is wholly determined by χ(2) since 2 generates the group of units modulo 9.
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
More generally, if m is an odd number the function
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.
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