\[\begin{align}\displaystyle\sum_{k=1}^{\infty} \dfrac{(-1)^{k+1} H_{k}}{k}\end{align}=\dfrac{1}{2} \left(\zeta(P)-\log^2 (Q)\right)\]

If the above equation is satisfied by positive integers \(P\) and \(Q\). Find \(P^Q\).

\(H_{n}\) is the \(n^{th}\) harmonic number.

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