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We prove two results, generalizing long existing knowledge regarding theclassical case of the Riemann zeta function and some of its generalizations.These are concerned with the question of Ingham who asked for optimal andexplicit order estimates for the error term $\Delta(x):=\psi(x)-x$, given anyzero-free region $D(\eta):=\{s=\sigma+it\in\mathbb{C}~:~ \sigma:=\Re s \ge1-\eta(t)\}$. In the classical case essentially sharp results are due to some40 years old work of Pintz. Here we consider a given a system of Beurlingprimes $\mathcal{P}$, the generated arithmetical semigroup $\mathcal{G}$ andthe corresponding integer counting function $N(x)$, and the corresponding errorterm $\Delta_{\mathcal{G}}(x):=\psi_{\mathcal{G}}(x)-x$ in the PNT of Beurling,where $\psi_{\mathcal{G}}(x)$ is the Beurling analog of $\psi(x)$. First weprove that if the Beurling zeta function $\zeta_{\mathcal{G}}$ does not vanishin $D(\eta)$, then the extension of Pintz' result holds:$|\Delta_{\mathcal{G}}(x)| \lex\exp(-(1-\varepsilon)\omega_\eta(x))~(x>x_0(\varepsilon))$, 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, thenanalogously to the classical case, $|\Delta_{\mathcal{G}}(x)| \gex\exp(-(1+\varepsilon)\omega_\eta(x))$ holds "infinitely often". This alsoshows that both main results are sharp apart from the arbitrarily small$\varepsilon>0$.

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