Short proof of the error bound in PNT assuming a zero-free strip?

Question

I am looking for a short proof of the fact that $\zeta(z)\neq 0$ for $\Re z>a$ implies the prime number theorem with an error bound $O(x^{a+\varepsilon})$ for any $\varepsilon>0$, which would be as elementary in spirit as Newman's proof of the PNT.

"Morally", it follows from the Mellin transform formula $$ \psi(x)=-\frac{1}{2\pi i}\int_{\Re s=\sigma}\frac{\zeta'(s)}{\zeta(s)}x^s \frac{ds}{s}=-\frac{x^\sigma}{2\pi}\int_{\mathbb{R}}\frac{\zeta'(\sigma+it)}{\zeta(\sigma+it)}e^{it\log x}\frac{dt}{\sigma+it}, $$

in which one then shifts the line of integration from $\sigma>1$ to $\sigma=a+\varepsilon$, picking up a pole at $s=1$ in the process. However, to rigorously do such shifting one seems to need (at least) estimates of $\zeta'/\zeta$ on the horizontal sides of the rectangle, which I don't see immediately how to obtain just from the non-vanishing of $\zeta$ and its simple properties.

The proofs I have seen in the literature (e. g. in the lecture notes of Elkies or in Edwards book) seem to use the Hadamard product representation of $\zeta$, the related series representation for $\zeta'/\zeta$, and the zero density estimates, and therefore, eventually, the functional equation.

Does an elementary proof of this fact (and, in particular, of the implication between the two forms of Riemann hypothesis) exist? Or is it known to be impossible, in the sense that it genuinely depends on the functional equation? E. g., are there convincing examples where the above "contour shifting" fails?

Answer 1

The "pretentious" Riemann hypothesis (RH) (which is in fact equivalent to the classical formulation of the RH) states that for all $\epsilon>0$, there exists a constant $c_{\epsilon}>0$ such that for every integer $k\geq 1$, we have

$\Big|\Big(\frac{\zeta'(s)}{\zeta(s)}+\frac{1}{s-1}\Big)^{(k)}\Big|\leq c_{\epsilon}k! 2^k (1+t^{\epsilon})$

uniformly for $s=\sigma+it$ with $1\leq\sigma<2$ and $0\leq t\leq e^k$ (see Section 17 of Granville). Using work of Koukoulopoulos, one can show that if the pretentious RH is true, then the error term in the prime number theorem is $O_{\epsilon}(x^{1/2+\epsilon})$ for all fixed $\epsilon>0$. I'm guessing that if $2^k$ is replaced by $\alpha^{-k}$, then the resulting pretentious quasi-RH suffices to obtain $O_{\epsilon}(x^{\alpha+\epsilon})$. Regardless of the value of $\alpha$, the transition from one statement does not appear to touch the functional equation. Whether you find the proof to be "in the spirit of Newman" is highly subjective and dependent on your personal goals; you should read Koukoulopoulos's beautiful paper and determine that for yourself. They are both largely "elementary" (depending on where you stand on the issue of Fourier/Mellin/Laplace inversion), but Newman uses (what I see as) very ad hoc analytic devices whereas Koukoulopoulos uses (what I see as) highly motivated sieve methods.