Can any interesting results concering prime numbers be obtained using the analytic properties of the polylogarithm, similar to how analytic methods are used on the zeta function to obtain results about primes?
$\text{Li}_s(x)$ is the polylogarithm, and $\zeta(s)$ is the zeta function: $$\text{Li}_s(x)=\sum_{n=1}^\infty\frac{x^n}{n^s} ,\text{ }\text{ } \text{ } \zeta(s)=\sum_{n=1}^\infty\frac{1}{n^s}$$ The zeta function satisfies $$-\frac{d}{ds}\ln(\zeta(s))=\sum_{p\text{ prime}}\frac{\ln(p)}{p^s-1}=\sum_{n=1}^\infty\frac{\Lambda(n)}{n^s}$$ and the polylogarithm satisfies a similar result: $$-\frac{d}{ds}\text{Li}_s(x)=\sum_{p\text{ prime}}\ln(p)( \frac{ \text {Li}_s(x^p)}{p^s}+\frac{ \text {Li}_s(x^{p^2})}{p^{2s}}+\frac{ \text {Li}_s(x^{p^3})}{p^{3s}}+....)=\sum_{n=1}^\infty\frac{\Lambda(n)\text{Li}_s(x^n)}{n^s}$$
Also, I don't know if its worth noting, but by substituting roots of unity for the argument of the polylogarithm, using some orthogonality relations and Möbius inversion one can obtain, $$-\frac{1}{a}\sum_{n=1}^\infty\frac{\mu(n)}{n^s}\sum_{t=0}^{a-1}\frac{d}{ds}\text{Li}_s(e^{2\pi itn/a})e^{-2\pi i t b/a}=\sum_{n=0}^\infty \frac{\Lambda(an+b)}{(an+b)^s}=\sum_{p\equiv \text{ b mod a }}\frac{\ln(p)}{p^s}+o(1)$$ And thus, $$-\frac{1}{a}\int_{s}^\infty\sum_{n=1}^\infty\frac{\mu(n)}{n^s}\sum_{t=0}^{a-1}\frac{d}{ds}\text{Li}_s(e^{2\pi itn/a})e^{-2\pi i t b/a}\ ds=\sum_{p\equiv \text{ b mod a }}\frac{1}{p^s}+o(1)$$
which I used to obtain results like: $$\sum_{p\equiv 2 \text{ mod 3}}\frac{\ln(p)p^s}{p^{2s}-1}=\frac{\zeta(s,\frac{2}{3})\zeta'(s,\frac{1}{3})-\zeta(s,\frac{1}{3})\zeta'(s,\frac{2}{3})}{\zeta(s,\frac{1}{3})^2-\zeta(s,\frac{2}{3})^2}$$ where $\zeta(s,q)$ are Hurwitz zeta functions. I know Dirichlet $L$ functions can be used to give similar results. I just thought it was worth mentioning, and in this case it's much easier to recognize divergence of the left hand sum as $\lim_{s\to 1}$.
