Proof. [02YD]

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Proof.

We have that $\displaystyle\pi\cot(\pi x)=\frac{1}{x}+O(1)$ for $x\to 0$ . Hence,

\operatorname{h}_{{\overline{{\mathcal{O}}(1)}}}(C_{r})=\sum_{j=1}^{\lfloor r/2\rfloor}\bigg(1-\frac{2\,j}{r+1}\bigg)\,\frac{j}{r+1}+O(r)=r\bigg(\sum_{j=1}^{\lfloor r/2\rfloor}\frac{1}{j}\bigg)+O(r)=r\log r+O(r).

∎