I have been trying to prove that $$\Re(\log\zeta(\sigma+it))=\sum_{n=2}^\infty\frac{\Lambda(n)}{n^\sigma\log n}\cos(t\log n),$$ but now I've given up, so I looked up the answer in the back of the book. It starts just the same way I was trying to go:
\begin{align*} \log\zeta(s)&=-\sum_p\log\left(1-\frac{1}{p^s}\right) \\ &=\sum_p\sum_{k=1}^\infty\frac{1}{kp^{ks}}. \end{align*}
But then the author just jumps right to the solution, not giving any intermediary step(s):
$$\sum_p\sum_{k=1}^\infty\frac{1}{kp^{ks}}=\sum_{n=1}^\infty\frac{\Lambda(n)}{n^\sigma\log n}\{\cos(t\log n)-i\sin(t\log n)\}.$$
Can someone please help me see what is going on in the last step?