I am having a trouble understanding some equality in the book "The Theory of the Riemann zeta-function".
On pp. 9, I read \[ \sum_{n \geq 1} \frac{\sigma_{a}(kn)}{n^{s}} = \prod_{p} \sum_{m \geq 0} \frac{\sigma_{a}(p^{l + m})}{p^{ms}}, \] where $k = \prod_{p}p^{l}$. But simply expanding on the right, it seems there are many terms that I can't find in the left series.
In addition, if the relation next to it, namely, \[ \sum_{n \geq 1} \frac{\sigma_{a}(kn)}{n^{s}} = \zeta(s) \zeta(s - a) \prod_{p|k} (\sum_{m \geq 0} \sigma_{a}(p^{l + m})/p^{ms}) / \sum_{m \geq 0}\sigma_{a}(p^{m})/p^{ms}) \] were correct, then I would have \[ \sum_{n \geq 1} \frac{\sigma_{a}(kn)}{n^{s}} = \prod_{p \quad not| k} ( \sum_{m \geq 0}\sigma_{a}(p^{m})/p^{ms}) \prod_{p|k} (\sum_{m \geq 0} \sigma_{a}(p^{l + m})/p^{ms}), \] but this seems to be hard to verify too.
An explanation on this part would be really appreciated.
Thanks.