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where the pattern for the negative signs is yet to be found. It would appear that if the power of $x$ is divisible by $2$ and the power of $x$ divided by $2$ is not equal to $1$, then the overall sign of the temr in the expansion is negative. Then the polynomials in the numerator must be investigated. It would appear that the polynomial coefficient of $2$ is negative if the power of $x$ is of the form $4n+2$, $n=0,1,2,3,...$. All other prime divisor polynomial coefficients appear to be positive. This then explains all terms (as far as we know).  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)^{q+1}p^*a^{D_m(q)-\frac{q}{p}}x^q (-1)^{q+1}p^*(q)a^{D_m(q)-\frac{q}{p}}x^q  \\ \prod_{k=1}^\infty 1+x^{p_k}=\exp\left(\sum_{q=2}^\infty \frac{1}{q}\sum_{p|q} (-1)^{q+1}p^*x^q\right) (-1)^{q+1}p^*(q)x^q\right)  \end{equation}  where $p^*$ $p^*(q)$  means \begin{equation} p^*=(-1)^{q/2}\cdot p^*(q)=(-1)^{q/2}\cdot  2,\; p=2\\ p^*=p,\; p^*(q)=p,\;  \mathrm{otherwise} \end{equation}