Lemma: Reciprocity Law for Floor Functions

Let $p,q\in\mathbb Z$ be odd and co-prime integers with $p > 2$ and $q > 2.$ Then the following closed formula for the sum of floor functions holds: $$\sum_{k=1}^{\frac{p-1}{2}}\left\lfloor\frac{kq}p\right\rfloor+\sum_{l=1}^{\frac{q-1}{2}}\left\lfloor\frac{lp}q\right\rfloor=\frac{p-1}{2}\cdot\frac{q-1}{2}.$$

Proofs: 1

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