Floor Function is Replicative

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Theorem

The floor function is a replicative function in the sense that:

$\displaystyle \forall n \in \Z_{> 0}: \sum_{k \mathop = 0}^{n - 1} \left \lfloor{x + \frac k n}\right \rfloor = \left \lfloor{n x}\right \rfloor$


Proof

\(\displaystyle \sum_{k \mathop = 0}^{n - 1} \left \lfloor{x + \dfrac k n}\right \rfloor\) \(=\) \(\displaystyle \left \lfloor{x n + \left \lfloor{x + 1}\right \rfloor \left({\left \lceil{n}\right \rceil - n}\right)}\right \rfloor\) Summation over $k$ of $\left \lfloor{x + \dfrac k y}\right \rfloor$, setting $y = n$
\(\displaystyle \) \(=\) \(\displaystyle \left \lfloor{x n + \left \lfloor{x + 1}\right \rfloor \left({n - n}\right)}\right \rfloor\) Real Number is Integer iff equals Ceiling
\(\displaystyle \) \(=\) \(\displaystyle \left \lfloor{n x + \left \lfloor{x + 1}\right \rfloor \times 0}\right \rfloor\)
\(\displaystyle \) \(=\) \(\displaystyle \left \lfloor{n x}\right \rfloor\)

$\blacksquare$


Historical Note

The fact that the Floor Function is Replicative was discovered by Charles Hermite, who published this result in $1884$.


Sources

  • 1884: Charles HermiteSur quelques conséquences arithmétiques des Formules de la théorie des fonctions elliptiques (Acta Math. Vol. 5: 297 – 330)