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

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 $O_{\varepsilon}(x^{1/2+\varepsilon}).$

EDIT January 13th 2014: I feel like I have to add a few details of what my ideas are presently. Intuitively, a non-optimal error term $O(g(x))$ for the pair $(F_{a},F_{s})$, i.e such that $F_{a}(x)=F_{s}(x)+O(g(x))$ and $\lim_{x\to\infty}\frac{F_{a}(x)-F_{s}(x)}{g(x)}=0$ can be expressed in an arbitrarily complicated way. Then conversely, the optimal error term $O(f(x))$ should be expressed as simply as possible.

We thus get the following conjecture:

Main conjecture: foreword

For a given possible error term $O(h(x))$ for the pair $(F_{a},F_{s})$, let's denote by $\mathbb{K}_{h}$ the minimal differential extension of $\mathbb{K}(x)$ $h(x)$ lies in. $O(h(x))$ will be called a "possible error term for the pair $(F_{a},F_{s})$ over $\mathbb{K}$". Let's say that such an $O(h(x))$ is "algebraically non-trivial" over $\mathbb{K}$ if and only if $h(x)\not\in\mathbb{K}(x)$.

Moreover, let's say that $O(f(x))$ is a quasi-optimal error term for the pair $(F_{a},F_{s})$ over $\mathbb{K}$ if and only if $F_{a}(x)=F_{s}(x)+O(f(x))$ and the quotient $\frac{F_{a}(x)-F_{s}(x)}{f(x)}$ is bounded but doesn't tend to $0$ as $x$ tends to $+\infty$.

Among all quasi-optimal error terms $O(f(x))$ for the pair $(F_{a},F_{s})$ there is an $O(f_{0}(x))$ such that $\mathbb{K}_{f_{0}}\subseteq\mathbb{K}_{f}$. Such an $O(f_{0}(x))$ will be called an optimal error term for the pair $(F_{a},F_{s})$ over $\mathbb{K}$.

Main conjecture: statement

Let $O(g(x))$ be any algebraically non-trivial possible error term for the pair $(F_{a},F_{s})$ over $\mathbb{K}$ and $O(f(x))$ be an optimal error term for the pair $(F_{a},F_{s})$ over $\mathbb{K}$. Then $\mathbb{K}_{f}\subseteq\mathbb{K}_{g}$.

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

EDIT April 28th 2014: heres comes a heuristics for the main conjecture. Generally speaking, the error term tends to infinity when the variable tends to infinity, so that when one tries to "get closer to optimality", one divides the already known error term by a function of $x$ that tends to $+\infty$ as $x$ tends to $\infty$. This can be done either dividing by a new function which lies in a possibly "bigger" extension of $\mathbb{K}(x)$, and the resulting expression is more complex (or as complex as it was), or dividing by a function of $x$ that tends to $+\infty$ as $x$ tends to $\infty$ already used in the given expression of the error term (hence in the same extension of $\mathbb{K}(x)$), and the expression gets more simple, so that the differential field generated by the resulting error term is either the given extension $\mathbb{K}(x)$ or a differential subfield of it. Once the error term is quasi-optimal, all you can do to get a truly optimal error term is dividing by a constant, hence the resulting differential field remains the same. This explains (vaguely, I admit) why the optimal error term has to be as simple as possible, and the corresponding differential field minimal. All that remains to be done is giving a really rigorous version of this argument.

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.


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