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Where $p_k$ is the $k^{th}$ prime.  Then \begin{equation}  \lim_{N\to\infty}(\log\Pi_N(a,x) - \log a^N) = \frac{2x^2}{2a}+\frac{3x^3}{3a} -\frac{2x^4}{4a^2} + \frac{5x^5}{5a} + \frac{(2-3a)x^6}{6a^3} +\frac{7x^7}{7a} -\frac{2x^8}{8a^4}+\frac{3x^9}{9a^3}+\frac{(2-5a^3)x^{10}}{10a^5}+\cdots  \end{equation} It would appear there are a few patterns here. \begin{itemize}  \item The power of $a$ in the denominator is the largest proper divisor of the power of $x$.  \item   \end{itemize}