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x^q \to \frac{1}{a^{D_m(q)}}\sum_{p|q} (-1)^?pa^{D_m(q)-\frac{q}{p}}  \end{equation}  where $D_m(q)$ is the largest proper divisor of $q$. This gives \begin{equation}  \lim_{N\to\infty}(\log\Pi_N(a,x) - \log a^N) = \sum_{q=2}^\infty \frac{1}{qa^{D_m(q)}}\sum_{p|q} (-1)^?pa^{D_m(q)-\frac{q}{p}}x^q \\  \prod_{k=1}^\infty 1+x^{p_k}=\sum_{q=2}^\infty \frac{1}{q}\sum_{p|q} (-1)^?px^q  \end{equation}