Prime power Gauss sums are zero

Question

Fix an odd prime $p$. Then for a positive integer $a$, I can look at the quadratic Legendre symbol Gauss sum

$$ G_p(a) = \sum_{n \,\bmod\, p} \left( \frac{n}{p} \right) e^{2 \pi i a n / p}$$

where I use $\left( \frac{\cdot}{\cdot} \right)$ to be the Legendre Symbol. We know how to calculate it explicitly, and that's all well and good.

Recently, I've had reason to consider a "prime power" Gauss Sum

$$G_{p^k}(a) = \sum_{n\,\bmod\,p^k}\left(\frac{n}{p^k}\right)e^{2\pi i a n/p^k}$$

and I've noticed that as long as $k > 1$, $G_{p^k}(a) = 0$.

A sketch of why this is true:

1. If $k$ is even, this is very easy to see as the Legendre symbol goes away, so we are summing the $p^k$th roots of unity.
2. If $k$ is odd, the way I see it is a bit less obvious: reduce the Legendre symbol to be $\left( \frac{n}{p} \right)$ so that it's $p$-periodic, write $n = pn' + n''$ fpr $0 \leq n' < p^{k-1}$ and $0 \leq n'' \leq p$. Reorder the sum to sum over the $p^{k-1}$ roots of unity first, which for $k > 1$ will still give $0$.

My question is this: there are many ways of viewing Gauss sums. They can be viewed as discrete Fourier transforms, generators of quadratic extensions in $\mathbb{Q(e^{2\pi i / p})}$, as eigenfunctions, natural character sums, etc. I suspect that there is some 'obvious' or 'clear' reason why these prime power Gauss sums should be zero.

Is there an obvious or clear reason why $G_{p^k}(a) = 0$?

Answer 1

What follows is essentially a way of rewording the argument you gave above.

Let $\chi$ be a Dirichlet character modulo $N$, and consider the Gauss sum $$\tau(\chi)=\sum_{k=1}^{N}\chi(k)e^{2\pi ik/N}.$$ Suppose that $q|N$ is the conductor of $\chi$ so that $\chi$ is induced by a primitive character modulo $q$. Call this primitive character $\chi^{\star}.$ Then Theorem $9.10$ of Montgomery and Vaughn states that $$\tau\left(\chi\right)=\mu\left(\frac{N}{q}\right)\chi^{\star}\left(\frac{N}{q}\right)\tau\left(\chi^{\star}\right).$$ When $k\geq2$ is even, the primitive character inducing $\left(\frac{n}{p^{k}}\right)$ is $\chi_{0},$ the principle character, and so $\frac{N}{q}=p^{k}.$ This is certainly not squarefree, so the $\mu\left(\frac{N}{q}\right)$ term is zero. Similarly, when $k\geq3$ is odd, the primitive character inducing $\left(\frac{n}{p^{k}}\right)$ is $\left(\frac{n}{p}\right),$ and so $\frac{N}{q}=p^{k-1}.$ Again this not squarefree, so the $\mu\left(\frac{N}{q}\right)$ term is zero implying that for $k\geq2$ $$\sum_{n=1}^{p^{k}}\left(\frac{n}{p^{k}}\right)e^{2\pi in/p^{k}}=0.$$