Advanced Number Theory Note #12: Separability property of Gauss sums associated with Dirichlet characters

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 \chi mod k, the Gauss sum associated with \chi is

G(n, \chi) = \sum_{m=1}^k \chi(m)e^{2\pi imn/k}

For the principal character mod k, usually denoted \chi_1, we have \chi_1(m) = 1 if gcd(m, k) = 1 and \chi_1(m) = 0 otherwise, and in this case the Gauss sum reduces to Ramanujan’s sum:

G(n, \chi_1) = c_k(n) = \sum_{\substack{m=1 \\ gcd(m, k)=1}}^k e^{2\pi imn/k}

Therefore the Gauss sums G(n, \chi) 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 \chi is any Dirichlet character mod k, then

G(n, \chi) = \overline{\chi}(n)G(1, \chi)      whenever     gcd(n, k) = 1

Remark: Note that if \chi is the principal character \chi_1 mod k, then this statement reduces to c_k(n) = \mu(k) if gcd(n, k) = 1. I mentioned in Advanced Number Theory Note #10 that the Möbius function as a sum of primitive kth 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 gcd(n, k) = 1 and suppose r runs through a complete residue system mod k. Then the numbers nr also run through a complete residue system mod k and since

|\chi(n)|^2 = \chi(n)\overline{\chi}(n) = 1

we have

\chi(r) = \chi(n)\overline{\chi}(n)\chi(r) = \overline{\chi}(n)\chi(nr)

Using this observation in the sum defining G(n, \chi) we get

G(n, \chi) = \sum_{r\hspace{1 mm} mod \hspace{1 mm}k}\chi(r)e^{2\pi inr/k}

= \overline{\chi}(n) \sum_{r\hspace{1 mm} mod \hspace{1 mm}k}\chi(nr)e^{2\pi inr/k}

= \overline{\chi}(n) \sum_{m\hspace{1 mm} mod \hspace{1 mm}k}\chi(m)e^{2\pi im/k}

= \overline{\chi}(n) G(1, \chi)

which proves the theorem. QED

Gauss sums G(n, \chi) which satisfy

G(n, \chi) = \overline{\chi}(n)G(1, \chi)               (1)

are said to be separable.

Example: As an example, let \chi be the (only) nonprincipal character mod 4. Then we have \chi(1) = 1, \chi(3) = -1, \chi(2) = \chi(4) = 0, so

G(n, \chi) = \chi(1)e^{2\pi in/4} + \chi(3)e^{2\pi i3n/4}

= i^n - (-i)^n = i^n(1 - (-1)^n)

Therefore

G(n, \chi) = 2i    if    n \equiv 1 (mod 4)

G(n, \chi) = -2i    if    n \equiv 3 (mod 4)

G(n, \chi) = 0    if    n \equiv 0 or 2 (mod 4)

But

G(1, \chi) = \chi(1)e^{2\pi i/4} + \chi(3)e^{2\pi i3/4} = 2i

so it is obvious that G(n, \chi) = \overline{\chi}(n)G(1, \chi) in each case. END

Theorem 1 says that G(n, \chi) is separable whenever n is relatively prime to the modulus k. The following theorem applies in the case of integers n which are not relatively prime to k:

Theorem 2. If \chi is a character mod k, the Gauss sum G(n, \chi) is separable for every n if and only if

G(n, \chi) = 0       whenever    gcd(n, k) > 1

Remark: In the context of the example above where \chi is the nonprincipal character mod 4, observe that we indeed do have G(n, \chi) = 0 if gcd(n, 4) > 1.

Proof of Theorem 2: By Theorem 1, separability always holds if gcd(n, k) = 1, but in the case gcd(n, k) > 1 we have \overline{\chi}(n) = 0 (by definition of Dirichlet characters) so equation (1) above holds if and only if G(n, \chi) = 0. QED

The next theorem gives a useful consequence of separability:

Theorem 3. If G(n, \chi) is separable for every n, then

|G(1, \chi)|^2 = k                       (2)

Remark: Again in the context of the example above where \chi is the nonprincipal character mod 4, observe that we indeed do have |G(1, \chi)|^2 = 4.

Proof of Theorem 3: We have

|G(1, \chi)|^2 = G(1, \chi)\overline{G(1, \chi)}

= G(1, \chi)\sum_{m=1}^k \overline{\chi}(m)e^{-2\pi im/k}

= \sum_{m=1}^k G(m, \chi)e^{-2\pi im/k}

= \sum_{m=1}^k \big(\sum_{r=1}^k \chi(r)e^{2\pi imr/k} \big) e^{-2\pi im/k}

= \sum_{r=1}^k \chi(r) \sum_{m=1}^k e^{2\pi im(r - 1)/k} = k\chi(1) = k

since the last sum over m is a geometric sum which vanishes unless r = 1. QED

By Theorem 2, separability of G(n, \chi) is equivalent to the vanishing of G(n, \chi) whenever gcd(n, k) > 1. It is natural to then ask: For which characters \chi will G(n, \chi) fail to vanish when gcd(n, k) > 1, 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 \chi be a Dirichlet character mod k and assume that G(n, \chi) \neq 0 for some n such that gcd(n, k) > 1. Then there exists a divisor d of k, d < k, such that

\chi(a) = 1    whenever     gcd(a, k) = 1    and    a \equiv 1 (mod d)              (3)

Proof: For the given n, let q = gcd(n, k) and let d = k/q. Then d|k and since q > 1 we have d < k. Choose any a satisfying gcd(a, k) = 1 and a \equiv 1 (mod d). Then we must have \chi(a) = 1 by the following argument.
Since gcd(a, k) = 1, in the sum defining G(n, \chi) we can replace the index of summation m by am and we get

G(n, \chi) = \sum_{m\hspace{1 mm} mod \hspace{1 mm}k}\chi(m)e^{2\pi inm/k}

= \sum_{m\hspace{1 mm} mod \hspace{1 mm}k}\chi(am)e^{2\pi inam/k}

= \chi(a) \sum_{m\hspace{1 mm} mod \hspace{1 mm}k}\chi(m)e^{2\pi inam/k}

Since a \equiv 1 (mod d) and d = k/q we can write

a = 1 + (bk/q)

for some integer b, and we then have

\frac{anm}{k} = \frac{nm}{k} + \frac{bknm}{qk} = \frac{nm}{k} + \frac{bnm}{q} \equiv \frac{nm}{k} (mod 1)

since q|n. To say that anm/k \equiv nm/k (mod 1) means simply that anm/k and nm/k differ by an integer, and hence we have

e^{2\pi ianm/k} = e^{2\pi i(nm/k + v)} = e^{2\pi inm/k}

so the sum for G(n, \chi) becomes

G(n, \chi) = \chi(a) \sum_{m\hspace{1 mm} mod \hspace{1 mm}k}\chi(m)e^{2\pi inm/k} = \chi(a)G(n, \chi)

Since G(n, \chi) \neq 0, this implies \chi(a) = 1 as claimed, so the theorem is proved. QED

Example: The following concrete example illustrates several of the ideas above. Suppose \chi is the nonprincipal character mod 6. The reduced residue classes mod 6 are \{1, 5\} 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 \chi which must have the square roots of unity as its values, so \chi(1) = 1, \chi(5) = -1. We will show that G(2, \chi) and G(4, \chi) are not separable, and |G(1, \chi)|^2 \neq 6. We have

G(2, \chi) = \chi(1)e^{2\pi i2/6} + \chi(5)e^{2\pi i5 \cdot 2/6}

= e^{2\pi i/3} - e^{4\pi i/3} = \sqrt{3}i \neq 0

and

G(4, \chi) = \chi(1)e^{2\pi i4/6} + \chi(5)e^{2\pi i5 \cdot 4/6}

= e^{4\pi i/3} - e^{2\pi i/3} = -\sqrt{3}i \neq 0

Therefore we have G(2, \chi) \neq 0 even though gcd(2, 6) = 2 > 1, and G(4, \chi) \neq 0 even though gcd(4, 6) = 2 > 1. It follows from Theorem 2 that neither G(2, \chi) nor G(4, \chi) can be separable. Furthermore, we have

G(1, \chi) = \chi(1)e^{2\pi i/6} + \chi(5)e^{2\pi i5/6}

= e^{\pi i/3} - e^{5\pi i/3} = \sqrt{3}i

and so

|G(1, \chi)|^2 = 3 \neq 6

To illustrate Theorem 4, observe that the proper divisors of k = 6 are 1, 2, 3. Now consider any a satisfying gcd(a, 6) = 1 and a \equiv 1 (mod 3). Then we must necessarily have a \equiv 1 (mod 6), so \chi(a) = \chi(1) = 1. Thus, d = 3 in the context of Theorem 4.
But note that we cannot have d = 2. A counterexample is the case a = 5. We have 5 \equiv 1 (mod 2) and gcd(5, 6) = 1, but \chi(5) = -1 \neq 1.
Similarly, we cannot have d = 1. A counterexample is again the case a = 5. We have 5 \equiv 1 (mod 1) and gcd(5, 6) = 1, but \chi(5) = -1 \neq 1.
Thus we cannot have either d = 1 or d = 2. We can only have d = 3 in the context of Theorem 4 in this case. END