deletions | additions       diff --git a/untitled.tex b/untitled.tex  index 944d975..a775723 100644  --- a/untitled.tex  +++ b/untitled.tex 
...
\prod_{k=1}^\infty 1+x^{p_k}=\exp\left(\sum_{q=2}^\infty \frac{1}{q}\sum_{p|q} (-1)^?px^q\right)  \end{equation}  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 \\  \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)  \end{equation}  where $p^*$ means \begin{equation}  p^*=-2, p=2\\  p^*=p, \mathrm{otherwise}  \end{equation}