Can the error term involved in the PNT be expressed in a Galois theoretic framework?

Question

According to Wikipedia, the current best error term for the prime number theorem is $\pi(x)-\mathrm{Li}(x)=O\left(x\exp\left(-\frac{A(\ln x)^{3/5}}{(\ln\ln x)^{1/5}}\right)\right)$, while RH is equivalent to $\pi(x)-\mathrm{Li}(x)=O(\sqrt{x}\ln x)$. Can these two error terms be expressed in a Galois theoretic framework, namely would the expression inside the big $O$ be an element of some extension of $\mathbb{Q}(x)$? If so, could we give a rigorous argument explaining why the error term under RH is in itself simpler than the unconditional one, hence giving further evidence towards the truth of RH?
Thanks in advance.