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Edit July 8th 2023: is the content of this question potentially formalizable in Lean so as to be made somewhat useful?

Edit February 21rst 2019 : through Galois correspondence, the minimality of $K_{h}$ should correspond to the maximality of the corresponding Galois group, hence providing a measure of the "niceness" of the optimal error term. Maybe the latter can be determined by requiring the maximal symmetry compatible with the considered problem.

Following 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 $F_{a}(x)=F_{s}(x)+O(f(x))$).

Following https://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 $F_{a}(x)=F_{s}(x)+O(f(x))$).