General Reciprocity Law

Theorem

 * $\ds \sum_{0 \mathop \le j \mathop < \alpha n} \map f {\floor {\dfrac {m j} n} } = \sum_{0 \mathop \le r \mathop < \alpha m} \ceiling {\dfrac {r n} m} \paren {\map f {r - 1} - \map f r} + \ceiling {\alpha n} \map f {\ceiling {\alpha m} - 1}$

for $\alpha \in \R$.