I was reading chapter $3$ proposition $3.2$ of Marius Overholt where I am stuck at some point where he uses the equation
$$\big|\prod_{p \leq x}\sum_{k=0}^\infty f(p^k)-\sum_{n\leq x}f(n)\big|=\big|\sum_{p_1,\cdots, p_r \leq x} f(p_1^{k_1})\cdots f(p_r^{k_r})-\sum_{n\leq x}f(n)\big|$$
I am not getting why is this true!! If you observe closely, you will get that in the R.H.S there is nothing about the $k_i$'s. If we assume this then everything falls very prominently. I am giving a snapshot of that page as well so that you could have a trace of the problem. If this is wrong then please give an alternate way to approach the proposition which could be found in that snapshot as well.
It is "almost" correct, but I think the meaning is clear.
Call $p_1,\ldots,p_r$ the primes $\le x$. Then \begin{align} \prod_{i=1}^r \sum_{k\ge 0}f(p_i^k)&=\left(1+f(p_1)+f(p_1^2)+\cdots\right)\left(1+f(p_2)+f(p_2^2)+\cdots\right)\cdots \\ &=\sum_{k_1 \ge 0} \cdots \sum_{k_r \ge 0} f(p_1^{k_1}) \cdots f(p_r^{k_r}). \end{align} In particular, just replace that $p_1,\ldots,p_r\le x$ with $k_1,\ldots,k_r \ge 0$.