A book I'm reading is asking me to prove the "formal [Euler's] identity for the zeta function". I'm unsure of what the word "formal" means here, as I thought it meant to prove the identity in the algebraic sense (similarly to e.g. formal power series), but I don't see an obvious algebraic formalization of the infinite sum $\sum_{n = 1}^{\infty} \frac{1}{n^s}$.
I looked it up online and found only an imprecise "algebraic" proof (essentially Euler's original proof which has some questionable manipulations of infinite sums) and a standard analytic bounding argument.
The word "formal" there just means "ignoring convergence issues". That's analogous to formal power series (and the notion of of a formal group). There is no need to try to develop a rigorous concept of formal Dirichlet series to solve your problem: just do the calculation without worrying about justifying the rearrangement of terms. That is a "formal calculation".
Elsewhere in math, "formal" can mean "quite rigorous", as in logic: see here. From that point of view, we should be calling formal power series "informal power series". :) But nobody does that.