prove for Jacobi symbols $\sum_{k=0}^{500}(\frac{k}{1001}) = 0$

Question

Could somebody help me prove that $\sum_{k=0}^{500}\left(\dfrac{k}{1001}\right) = 0$? I think that there might be a bijection with $\sum_{k=0}^{500}\left(\dfrac{k}{501}\right)$ but I don't know how to prove this

Answer 1

It is well known that $S_m = \sum_{k = 0}^{m-1} \left(\dfrac{k}{m}\right) = 0$, write the sum as $S_{1001} = \sum_{k = 0}^{500} \left(\dfrac{k}{1001}\right) + \sum_{k = 0}^{500} \left(\dfrac{1001 - k}{1001}\right) = 2\sum_{k = 0}^{500} \left(\dfrac{k}{1001}\right)$, therefore $\sum_{k = 0}^{500} \left(\dfrac{k}{1001}\right) = 0$

Answer 2

Use the fact that $\left(\frac{a}{q}\right)=\left(\frac{b}{q}\right)$, whenever $a\equiv b\pmod q$.