Let $p$ be an odd prime, $v \in \mathbb{N}$ be a positive integer, and $c\in \mathbb{Z}$. Set \begin{align} G(c,p^v):=\sum_{\substack{d \bmod p^v \\ (d,p^v)=1}}{ \left(\frac{d}{p^v}\right) {e}^{ { \frac{2\pi i cd}{p^v} }}},&&\text{where }\left(\frac{\cdot}{p}\right) \text{ is the usual Legendre symbol}. \end{align}
My question: What is $$ \sum_{i=0}^{v}G(c,p^i) ~~? $$
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