A 1950 result of Tur'an establishes an equivalence between any prime number theorem of the form $\operatorname{li}(x)-\pi(x)= O(xe^{-C(\log x)^\alpha}) \ (x \to \infty)$ and a certain class of zero-free regions of $\zeta(s)$. See: P. Tur'an, On the remainder-term of the prime-number formula, II, Acta Math. Acad. Sci. Hungar. 1 (1950) 155--166.


Has this equivalence been extended to prime number theorems of the form $\operatorname{li}(x)-\pi(x)= O(xe^{-C(\log x)^\alpha(\log \log x)^\beta}) \ (x \to \infty)$? If so, what is the most general known equivalence of this type?



Zero-free regions of $\zeta(s)$ equivalent to prime number theorems with error bound
Yes. All of the results quoted in the answer are stated in Pintz's paper "[On the remainder term of the prime number formula. II. On a theorem of Ingham.](https://zbmath.org/0447.10038)" (Acta Arith. 37, 209-220 (1980)). Below, your $(\alpha,\beta)$ correspond to my $(1/(a+1),b/(a+1))$.


1. One direction is older than Turán and goes back to Ingham. He established that if $\zeta(s)$ does not vanish when $\Re s>1-\eta(|\Im s|)$, and $\eta >0$ is a continuously-differentiable decreasing function satisfying some mild conditions, then one has $$\pi(x)-\mathrm{Li}(x)\ll\_{\varepsilon} xe^{-\frac{1}{2}(1-\varepsilon)\omega(x)}$$ for
$$\omega(x) := \min\_{t \ge 1} (\eta(t)\log x+\log t)=\min\_{v \ge 0} (\eta(e^v)\log x+ v).$$
In particular, this applies to $\eta(t) = c(\log t)^{-a}(\log \log t)^{b}$ ($a>0$), which gives $\omega(x) \sim d\_{c,a,b}(\log x)^{\frac{1}{a+1}} (\log \log x)^{\frac{b}{a+1}}$ by a short computation.
This is Theorem 22 in Ingham's book "[The distribution of prime numbers](https://zbmath.org/0006.39701)" (1932). The formulation above is from Pintz's paper. This resolves one direction of your question. Pintz was interested in optimal exponents, and in Theorem 1 he sharpened Ingham's result, proving, in the same notation as above, that
$$\pi(x)-\mathrm{Li}(x)\ll\_{\varepsilon} \frac{x}{\log x}e^{-(1-\varepsilon)\omega(x)}.$$
Moreover, he doesn't require $\eta$ to be differentiable, it can be any continuous decreasing function taking values in $(0,1/2]$. (In particular, the mild conditions of Ingham, which include $1/\eta(t) = O(\log t)$, are not needed, and any choice of $a>0$ and $b \in \mathbb{R}$ is allowed.)
2. [Turán](https://zbmath.org/0041.37102) and [Staś](https://zbmath.org/0104.27001) studied converses to Ingham's result. As you are already familiar with Turán's work, let me state Staś's as it appears in Pintz. If $\eta> 0$ is a continuous decreasing function satisfying some mild conditions, and
$$|\psi(x)-x|\le c xe^{-\frac{1}{2}(1-\varepsilon)\omega(x)}$$
for the same $\omega$ as before, then $\zeta(s) \neq 0$ for
$$\Re s > 1-\frac{1-\varepsilon}{1600}\eta(|\Im s|^{4/(1-\varepsilon)}), \qquad |\Im s| \gg\_{\varepsilon} 1.$$
Again, this applies to $\eta(t) = c(\log t)^{-a}(\log \log t)^{b}$, for which $\omega(x) \sim d\_{c,a,b}(\log x)^{\frac{1}{a+1}} (\log \log x)^{\frac{b}{a+1}}$. This shows that if $|\psi(x)-x| \ll x e^{-c'(\log x)^{\frac{1}{a+1}} (\log \log x)^{\frac{b}{a+1}}}$ then $\zeta(s) \neq 0$ for $\Re s > 1-c''(\log t)^{-a}(\log \log t)^{b}$.
This gives the second direction of your question and might satisfy you already. Although you ask about $\psi(x)-x$ and not $\pi(x)-\mathrm{Li}(x)$, this difference is just a matter of partial summation.
3. A sharper version of Staś's result is Theorem 2 of Pintz (which has slightly different assumptions). The theorem says that if there is an infinite sequence of zeroes of $\zeta$ with $\Re s > 1-\eta(|\Im s|)$, where $\eta>0$ is a continuous decreasing function satisfying some mild conditions, then $$\pi(x)-\mathrm{Li}(x) = \Omega\_{\pm}(x e^{-(1+\varepsilon)\omega(x)}).$$
This might not be as useful as Staś's work due to the requirement of infinitely many zeros. However, see Theorem 2' of Pintz.


