  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 $\Delta(x):=\psi(x)-x$, given any
zero-free region $D(\eta):=\{s=\sigma+it\in\mathbb{C}~:~ \sigma:=\Re s \ge
1-\eta(t)\}$. 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 $\Delta_{\mathcal{G}}(x):=\psi_{\mathcal{G}}(x)-x$ 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:
$|\Delta_{\mathcal{G}}(x)| \le
x\exp(-(1-\varepsilon)\omega_\eta(x))~(x>x_0(\varepsilon))$, 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, then
analogously to the classical case, $|\Delta_{\mathcal{G}}(x)| \ge
x\exp(-(1+\varepsilon)\omega_\eta(x))$ holds "infinitely often". This also
shows that both main results are sharp apart from the arbitrarily small
$\varepsilon>0$.
