In a previous note I studied a multiplicative property of quadratic Gauss sums. (I intend to use this result in a future note which will explore in detail a proof of the law of quadratic reciprocity using Cauchy’s residue theorem from Complex Analysis). Quadratic Gauss sums are a special case of Gauss sums associated with Dirichlet characters, and in the present note I want to explore some important properties of the latter, particularly a separability property which plays a role in other areas I want to look into in future notes. As usual I will try to illustrate key abstract ideas with concrete examples.
For any Dirichlet character mod , the Gauss sum associated with is
For the principal character mod , usually denoted , we have if and otherwise, and in this case the Gauss sum reduces to Ramanujan’s sum:
Therefore the Gauss sums can be viewed as a generalisation of Ramanujan’s sums.
The following is the key separability property of Gauss sums which is of interest here:
Theorem 1. If is any Dirichlet character mod , then
Remark: Note that if is the principal character mod , then this statement reduces to if . I mentioned in Advanced Number Theory Note #10 that the Möbius function as a sum of primitive th roots of unity is an interesting special case of Ramanujan’s sums and Gauss sums, and this connection between the three is made beautifully clear here.
Proof of Theorem 1: Let and suppose runs through a complete residue system mod . Then the numbers also run through a complete residue system mod and since
Using this observation in the sum defining we get
which proves the theorem. QED
Gauss sums which satisfy
are said to be separable.
Example: As an example, let be the (only) nonprincipal character mod 4. Then we have , , , so
if (mod 4)
if (mod 4)
if or (mod 4)
so it is obvious that in each case. END
Theorem 1 says that is separable whenever is relatively prime to the modulus . The following theorem applies in the case of integers which are not relatively prime to :
Theorem 2. If is a character mod , the Gauss sum is separable for every if and only if
Remark: In the context of the example above where is the nonprincipal character mod 4, observe that we indeed do have if .
Proof of Theorem 2: By Theorem 1, separability always holds if , but in the case we have (by definition of Dirichlet characters) so equation (1) above holds if and only if . QED
The next theorem gives a useful consequence of separability:
Theorem 3. If is separable for every , then
Remark: Again in the context of the example above where is the nonprincipal character mod 4, observe that we indeed do have .
Proof of Theorem 3: We have
since the last sum over is a geometric sum which vanishes unless . QED
By Theorem 2, separability of is equivalent to the vanishing of whenever . It is natural to then ask: For which characters will fail to vanish when , and thus be nonseparable? The following (final) theorem addresses this question, and is one which leads to a whole subfield concerned with induced moduli and induced Dirichlet characters which I will explore in detail in a later note.
Theorem 4. Let be a Dirichlet character mod and assume that for some such that . Then there exists a divisor of , , such that
whenever and (mod ) (3)
Proof: For the given , let and let . Then and since we have . Choose any satisfying and (mod ). Then we must have by the following argument.
Since , in the sum defining we can replace the index of summation by and we get
Since (mod ) and we can write
for some integer , and we then have
since . To say that (mod 1) means simply that and differ by an integer, and hence we have
so the sum for becomes
Since , this implies as claimed, so the theorem is proved. QED
Example: The following concrete example illustrates several of the ideas above. 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 , . We will show that and are not separable, and . We have
Therefore we have even though , and even though . It follows from Theorem 2 that neither nor can be separable. Furthermore, we have
To illustrate Theorem 4, observe that the proper divisors of are , , . Now consider any satisfying and (mod 3). Then we must necessarily have (mod 6), so . Thus, in the context of Theorem 4.
But note that we cannot have . A counterexample is the case . We have (mod 2) and , but .
Similarly, we cannot have . A counterexample is again the case . We have (mod 1) and , but .
Thus we cannot have either or . We can only have in the context of Theorem 4 in this case. END