I am doing a project on the Riemann-Zeta Function which begins by examining the Euler Product Formula. I understand the proof up until the point where it is made 'rigorous'. In other words, I understand the use of the fundamental theorem of arithmetic to show that $$\prod_{p\text{ prime}} \frac{1}{1-\frac{1}{p^s}}$$
is equal to a sum of terms of the form
$$\frac{1}{(p_{1}^{n_{1}}p_{2}^{n_{2}}p_{3}^{n_{3}} \cdots p_{r}^{n_{r}})^s}$$
for distinct primes $p_i$ and natural numbers $n_i$. This, to me, completes the proof that $$\sum_{n=1}^{\infty} \frac{1}{n^s} = \prod_{p\text{ prime}} \frac{1}{1-\frac{1}{p^s}}$$
However, in rigorous proofs, there is talk of an 'error term' using the Big O function or something similar. I don't understand why this is necessery, what it shows, and how it is showing it. Any help is greatly appreciated.