The method of Pintz for the Ingham question about the connection of distribution of $\zeta$-zeros and order of the error in the PNT in the Beurling context

We prove two results, generalizing long existing knowledge regarding the classical case of the Riemann zeta function and some of its generalizations. These are concerned with the question of Ingham who asked for optimal and explicit order estimates for the error term $\mathrm{\Delta }(x):=\psi (x)-x$, given any zero-free region $D(\eta ):=\{s=\sigma +it\in \mathbb{C}:\sigma :=\mathrm{\Re }s\ge 1-\eta (t)\}$. In the classical case essentially sharp results are due to some 40 years old work of Pintz. Here we consider a given a system of Beurling primes $\mathcal{P}$, the generated arithmetical semigroup $\mathcal{G}$ and the corresponding integer counting function $N(x)$, and the corresponding error term ${\mathrm{\Delta }}_{\mathcal{G}}(x):={\psi }_{\mathcal{G}}(x)-x$ in the PNT of Beurling, where ${\psi }_{\mathcal{G}}(x)$ is the Beurling analog of $\psi (x)$. First we prove that if the Beurling zeta function ${\zeta }_{\mathcal{G}}$ does not vanish in $D(\eta )$, then the extension of Pintz’ result holds: $\mathrm{\mid }{\mathrm{\Delta }}_{\mathcal{G}}(x)\mathrm{\mid }\le x\exp (-(1-\epsilon ){\omega }_{\eta }(x))(x>x_0(\epsilon ))$, where ${\omega }_{\eta }(y)$ is the naturally occurring conjugate function to $\eta (t)$, introduced into the field by Ingham. In the second part we prove a converse: if ${\zeta }_{\mathcal{G}}$ has an infinitude of zeroes in the given domain, then analogously to the classical case, $\mathrm{\mid }{\mathrm{\Delta }}_{\mathcal{G}}(x)\mathrm{\mid }\ge x\exp (-(1+\epsilon ){\omega }_{\eta }(x))$ holds “infinitely often”. This also shows that both main results are sharp apart from the arbitrarily small $\epsilon >0$.