Follwing http://math.stackexchange.com/questions/635659/can-the-error-term-involved-in-the-pnt-be-expressed-in-a-galois-theoretic-framew?noredirect=1#comment1341143_635659, I vainly tried to find references about what I would call "differential Galois number theory", which would consist in using techniques arising from differential fields and maybe differential Galois theory to obtain an a priori form for the best possible error term in analytic number theory problems (that is, the "simplest" function $f$ such that $\frac{F_{a}(x)-F_{s}(x)}{f(x)}$ is bounded but doesn't tend to $0$ as $x$ tends to $+\infty$, where $F_{a}(x)$ is an arithmetic function and $F_{s}(x)$ a "smooth" function such that $F_{a}(x)=F_{s}(x)+O(f(x))$).

My goal is to formalize the idea that, most of the time, error terms under big conjectures such as (G)RH and so on appear to be "nicer" than what we manage to prove unconditionally, and maybe to give further evidence for such conjectures. Maybe it would also be possible to establish a link between the symmetries of the problem involving a given arithmetic function $F_{a}(x)$ and the considered minimal differential extension of, say, $\mathbb{C}(x)$ its "real" error term $f(x)$ lies in. One can also expect to get explicit constants instead of rather inaccurate error terms like $O_{\varepsilon}(x^{1/2+\varepsilon}).$

Does someone know whether such ideas have been considered so far? If so, could I get a few references?
Thanks in advance.