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 $\mathrm{\Delta }(\mathit{x}):=\mathit{\psi }(\mathit{x})-\mathit{x}$, given any zero-free region $\mathcal{D}(\mathit{\eta }):=\{\mathit{s}=\mathit{\sigma }+\mathit{i}\mathit{t}\in \mathbb{C}:\mathit{\sigma }:=\mathrm{\Re }\mathit{s}\ge 1-\mathit{\eta }(\mathit{t})\}$. In the classical case essentially sharp results are due to some 40 years old work of Pintz.

Here we consider a given system of Beurling primes $\mathcal{P}$, the generated arithmetical semigroup $\mathcal{G}$, the corresponding integer counting function $\mathcal{N}(\mathit{x})$, and the respective error term ${\mathrm{\Delta }}_{\mathcal{P}}(\mathit{x}):={\mathit{\psi }}_{\mathcal{P}}(\mathit{x})-\mathit{x}$ in the PNT of Beurling, where ${\mathit{\psi }}_{\mathcal{P}}(\mathit{x})$ is the Beurling analogue of $\mathit{\psi }(\mathit{x})$. First we prove that if the Beurling zeta function ${\mathit{\zeta }}_{\mathcal{P}}$ does not vanish in $\mathcal{D}(\mathit{\eta })$, then the extension of Pintz’ result holds: $|{\mathrm{\Delta }}_{\mathcal{P}}(\mathit{x})|\le \mathit{x}exp((1+\mathit{\epsilon }){\mathit{\omega }}_{\mathit{\eta }}(log\mathit{x}))(\mathit{x}>{\mathit{x}}_0(\mathit{\epsilon }))$, where ${\mathit{\omega }}_{\mathit{\eta }}(\mathit{y}):=\mathcal{L}(\mathit{\eta })(\mathit{y})$ is the naturally occurring conjugate function—essentially the Legendre transform—of $\mathit{\eta }(\mathit{t})$, introduced into the field by Ingham. In the second part we prove a converse: if ${\mathit{\zeta }}_{\mathcal{P}}$ has an infinitude of zeroes in the given domain, then analogously to the classical case, $|{\mathrm{\Delta }}_{\mathcal{P}}(\mathit{x})|\ge \mathit{x}exp((1-\mathit{\epsilon }){\mathit{\omega }}_{\mathit{\eta }}(log\mathit{x}))$ holds “infinitely often”. This also shows that both main results are sharp apart from the arbitrarily small $\mathit{\epsilon }>0$.

The classical results of Pintz used many facts about the Riemann zeta function. Recently we worked out a number of analogous results—including some construction of quasi-optimal integration paths, a Riemann–von Mangoldt-type formula, a Carlson-type density theorem, and a Turán-type local density theorem—for the Beurling context, too. These, together with Turán’s power sum theory, all play some indispensable role in deriving the main results of the paper.