In this (rather long) note, intended mainly as a technical memo for myself, I want to explore in detail the concept of a primitive Dirichlet character mod k, and the role of induced moduli in determining primitive and non-primitive Dirichlet characters. (I plan to expand further on some aspects of primitive characters in a couple of subsequent notes). As usual I will try herein to illustrate key ideas with concrete examples.
Dirichlet characters form a group and satisfy other strong properties, such as being completely multiplicative, which make them mathematically interesting in their own right as well as useful. (See my Note about them). For example, one of their most important roles is in Dirichlet L-functions which in turn are used in the proof of the prime number theorem, something I will explore in great detail in future notes. Primitive Dirichlet characters in particular are important because all Dirichlet characters can be viewed as extensions of primitive ones. The primitive Dirichlet characters are the irreducible ones from which the others are made, somewhat analogously to how prime numbers are the irreducible integers from which other integers are made.
Primitive Dirichlet characters are defined using the concept of an induced modulus which I have already used in a couple of earlier notes. In one previous note I tried to formulate an intuitively simple proof of a result concerning Dirichlet characters with two induced moduli. The concept of an induced modulus was also central to the separability of Gauss sums which I explored in Advanced Number Theory Note #12. I briefly mentioned its relevance in the preamble to Theorem 4 there, and the basic definition of an induced modulus is actually contained in equation (3) in Theorem 4 of that note.
Definition of induced modulus. Let be a Dirichlet character mod and let be any positive divisor of . The number is called an induced modulus for if we have
whenever and (mod ) (1)
Example: Suppose is the nonprincipal character mod 6. The reduced residue classes mod 6 are so there are two Dirichlet characters mod 6. One is the principal character taking the value 1 for both 1 and 5. The other is the nonprincipal character which must have the square roots of unity as its values, so , . The proper divisors of are , , . Now consider any satisfying and (mod 3). Then we must necessarily have (mod 6), so . Thus, is an induced modulus of . But note that neither nor are induced moduli. A counterexample for both is the case . We have (mod 1) and , but . Similarly, we have (mod 2) and , but . Thus, the nonprincipal character mod 6 has only one induced modulus smaller than 6. END
In the above example the divisor 1 of 6 was found not to be an induced modulus of the nonprincipal character mod 6. The following theorem shows that there is only one character for each mod k such that 1 is an induced modulus, and that is the principal character.
Theorem 1. Let be a Dirichlet character mod . Then 1 is an induced modulus for if and only if is the principal character .
Proof: Suppose first that . Then for all which are relatively prime to . But since all such integers also satisfy (mod 1), the number 1 is an induced modulus. Conversely, suppose that the number 1 is an induced modulus. Then whenever and so it must be the case that since both and vanish when the argument is not coprime with . QED
Observe that for any Dirichlet character mod the modulus itself is an induced modulus, because is a divisor of and whenever and (mod ). The Dirichlet character is called primitive when it has no other induced moduli. Formally:
Definition of primitive characters. A Dirichlet character mod is said to be primitive mod if it has no induced modulus . Thus, a primitive Dirichlet character mod will be such that, for every proper divisor of , there is some integer (mod ) such that and .
Note that for any modulus , the principal character will not be primitive since it will have as an induced modulus. However, as the next theorem shows, if the modulus is a prime number, then every nonprincipal character mod is primitive.
Theorem 2. Every nonprincipal character modulo a prime is a primitive character mod .
Proof: The only proper divisor of is 1, but since is not a principal character the number 1 cannot be an induced modulus, so must be primitive since it has no induced modulus . QED
The next theorem establishes an important and useful property of Dirichlet characters, namely that is an induced modulus for mod if and only if is periodic mod on those integers relatively prime to . This can sometimes be used to identify induced moduli very quickly by looking at the pattern of a Dirichlet character’s values (see examples below).
Theorem 3. Let be a Dirichlet character mod and assume with . Then is an induced modulus for if and only if
whenever and (mod ) (2)
Proof: Observe first that if (2) holds, then simply setting gives equation (1), so must be an induced modulus if (2) holds. Going the other way, suppose that is an induced modulus. We will prove that then (2) must hold. Choose and such that
and (mod )
The following argument then shows . Let be the reciprocal of mod , so that
This reciprocal exists because . And note that also , so then we must have . Since , we can deduce immediately that
and therefore it must be the case that since is an induced modulus. But since (mod ) we deduce that
But since , so we can cancel to deduce that which is what we needed to prove. QED
The following examples illustrate the situation.
Example: The following table shows the values of one of the Dirichlet characters mod 9.
We can see straight away that this table is periodic modulo 3 so by Theorem 3 it must be the case that 3 is an induced modulus for . In fact, acts like the following Dirichlet character
in the sense that for all . Here, is a primitive Dirichlet character and is a non-primitive character called an extension of .
Whenever is an extension of a character modulo , then will be an induced modulus for because we will necessarily have whenever and (mod ). In the above case we have for example with and (mod ). END
Example: Here is an example showing that it is possible for a character mod to have an induced modulus without being an extension of any character mod . The case is the one considered in the first example above, where is the (only) nonprincipal character mod 6.
As seen earlier, in this case the number 3 is an induced modulus because whenever and (mod ). (There is only one such in this case, namely ). However, is not an extension of any character modulo 3, because the only characters modulo 3 are the nonprincipal one shown above and the principal character
Since , we see immediately that cannot be an extension of either or . END
Example: There are four reduced residue classes mod 8, namely 1, 3, 5, 7, and thus four Dirichlet characters mod 8 as follows.
We look for induced moduli among the proper divisors of 8 other than 1, namely 2, 4. We see immediately that , and are counterexamples for 2 being an induced modulus in the cases , and . Also and are counterexamples for 4 being an induced modulus in the cases and . However, it is immediately apparent that is periodic mod 4, so 4 must be an induced modulus for . END
Example: There are four reduced residue classes mod 12, namely 1, 5, 7, 11, and thus four Dirichlet characters mod 12 as follows.
Again we look for induced moduli among the proper divisors of 12 other than 1, namely 2, 3, 4, 6. We see immediately that , and are counterexamples for 2 being an induced modulus in the cases , and .
Also and are counterexamples for 4 being an induced modulus in the cases and . However, whenever and (mod ) we have so 4 is an induced modulus for .
and are counterexamples for 3 and 6 being induced moduli in the cases , . However, whenever and (mod ) we have so 3 is an induced modulus for . We also see that is periodic mod 6, so it follows from Theorem 3 that 6 must be an induced modulus for . END
The above examples also help to clarify the following theorem.
Theorem 4. Let be a Dirichlet character modulo , and assume and . Then the following two statements are equivalent:
(a). is an induced modulus for
(b). There is a character modulo such that
for all (3)
where is the principal character modulo .
Example: In examples above we saw that 4 is an induced modulus for mod 8 as well as mod 12. Therefore there must be a character modulo 4 such that
in both cases. This character is the only nonprincipal character mod 4, which takes the values and .
Similarly, we saw that 3 is an induced modulus for mod 12, so there must be a character modulo 3 such that
This character is the only nonprincipal character mod 3, which takes the values and .
Finally, we saw that 6 is an induced modulus for mod 12, so there must be a character modulo 6 such that
This character is the only nonprincipal character mod 6, which takes the values and . END
Proof of Theorem 4: First, we assume (b) holds and then show that (a) must be true. If (b) holds we can choose any satisfying and (mod ) and for this we will then have and also (because by assumption is a character mod , and (mod )). Therefore we will have for any such and so is an induced modulus, which proves (a).
Going the other way, we now assume that (a) holds and try to find a character modulo for which equation (3) above holds. We can obtain the desired character by defining as follows. If , let . In this case we also have (because ) so equation (3) holds because both sides are zero.
If then there exists an integer such that
(mod ) and
[This can be proved immediately with Dirichlet’s theorem on primes in arithmetic progressions. (See my Facebook Note for a detailed discussion of Dirichlet’s theorem and its proof). The arithmetic progression contains infinitely many primes. Simply choose one that does not divide and call it ].
Having chosen , which is unique modulo , we then define
The number is then a well defined character mod . It takes equal nonzero values at numbers which are congruent mod and coprime with , and takes the value zero by definition at numbers which are not coprime with .
We now verify that equation (3) holds for all . If , then it must also be the case that , so for some (mod ). So we have , , and (mod ) and since is an induced modulus by (a) it then follows from Theorem 3 above that
where the last equality holds since for all . This confirms that (b) holds in the case .
If , then and both sides of (3) are zero. This confirms that (b) holds in the case , so (b) holds for all if (a) is true. QED
It is useful to introduce the following terminology at this stage.
Definition. Let be a Dirichlet character mod . The smallest induced modulus for is called the conductor of .
The following theorem finally establishes the role that primitive Dirichlet characters play as the building blocks of Dirichlet characters generally.
Theorem 5. Every Dirichlet character mod can be expressed as a product
for all (4)
where is the principal character mod and is a primitive character modulo the conductor of .
Proof: Let be the conductor of . We already know from Theorem 4 that can be expressed as a product of the form (4), where is a character mod . To prove Theorem 5 we therefore only need to prove the additional claim that is primitive mod .
The proof will be by contradiction, assuming that is not primitive mod . If is not primitive mod there is a proper divisor of which is an induced modulus for . In equation (4) choose any such that
and (mod )
Then we will also have (since ) and since is an induced modulus for we must have . But then equation (4) says
so is also an induced modulus for which is smaller than . This is a contradiction since is supposed to be the conductor of . Therefore the assumption that is not primitive mod in (4) must be false. QED