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 $\alpha \in \R$.

Proof

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Sources

• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.4$: Integer Functions and Elementary Number Theory: Exercise $46$