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.
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 χ(a)=1 for a coprime to n and 1 mod d: equivalently, χ(a) = χ(b) whenever a, b are congruent mod d and each coprime to n. A character is primitive if there is no smaller induced modulus.
We can formalise differently this by defining characters χ1 mod N1 and χ2 mod N2 to be co-trained if for some modulus N such that N1 and N2 both divide N we have χ1(n) = χ2(n) for all n coprime to N: that is, there is some character χ* induced by each of χ1 and χ2. 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 L-functions.
Read more about this topic: Dirichlet Character
Famous quotes containing the words primitive, characters and/or conductor:
“The glory of the farmer is that, in the division of labors, it is his part to create. All trade rests at last on his primitive activity.”
—Ralph Waldo Emerson (1803–1882)
“His leanings were strictly lyrical, descriptions of nature and emotions came to him with surprising facility, but on the other hand he had a lot of trouble with routine items, such as, for instance, the opening and closing of doors, or shaking hands when there were numerous characters in a room, and one person or two persons saluted many people.”
—Vladimir Nabokov (1899–1977)
“I was the conductor of the Underground Railroad for eight years, and I can say what most conductors can’t say—I never ran my train off the track and I never lost a passenger.”
—Harriet Tubman (1821–1913)