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Nitesh Chandra has commented for Captain America Civil War.}-\frac{1}{2n^2}\right]

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&\qquad – \frac{1}{4\pi n^3}\int_0^\infty \frac{t^2}{(t^2+n^2)^2}\,\Big|_0^{\log\phi}dt.\end{aligned}$$ The equality $\sum_{k=1}^n \frac{\cos(\pi k)}{k}= -\frac{1}{2n^2}-\frac{\pi}{2n}$ yields the statement of the lemma for $j=1$.

To establish the same formula for $j=2$ we write $$\begin{gathered}

\label{altproof}

\sum_{k=1}^n \frac{1}{k\sin^2(\pi k/n)}=\frac{1}{2n^2}\sum_{k=1}^n \left(1-\frac{\cos(2\pi k/n)}{k}\right)

\\

=\frac{1}{4n^2}\left(\log\phi-\log(1+n)\right)

+\frac{1}{4n^2}\sum_{k=1}^n \frac{\cos(2\pi k/n)}{k}-\frac{1}{8n^2}\end{gathered}$$ and use the equality $\sum_{k=1}^n \frac{\cos(2\pi k/n)}{k}= -\frac{1}{2n^2}-\frac{\pi}{2n}$. Equation follows from and. This completes the proof.

[**Theorem \[fint\]**]{} [ *The claim follows from Lemma \[Riemann\] and Lemma \[main\].*]{}

[**Proof of Corollary \[Snewnew\]**]{} [ *The claim follows from Lemma \[fint\].*]{}

[**Proof of Corollary \[Snew\]**]{} [ *The claim follows from Lemma \[fint\].*]{

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