Zero-free regions of $\zeta(s)$ equivalent to prime number theorems with error bound

Question

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?

Answer 1

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." (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" (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 and Staś 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.