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Continuing previous study of the Beurling zeta function, here we prove tworesults, generalizing long existing knowledge regarding the classical case ofthe Riemann zeta function and some of its generalizations.  First, we address the question of Littlewood, who asked for explicitoscillation results provided a zeta-zero is known. We prove that given a zero$\rho_0$ of the Beurling zeta function $\zeta_P$ for a given number systemgenerated by the primes $P$, the corresponding error term$\Delta_P(x):=\psi_{P}(x)-x$, where $\psi_{P}(x)$ is the von Mangoldt summatoryfunction shows oscillation in any large enough interval, as large as$(\pi/2-\varepsilon) x^{\Re \rho_0}/|\rho_0|$.  The somewhat mysterious appearance of the constant $\pi/2$ is explained inthe study. Finally, we prove as the next main result of the paper thefollowing: given $\varepsilon>0$, there exists a Beurling number system withprimes $P$, such that $|\Delta_P(x)| \le (\pi/2+\varepsilon)x^{\Re\rho_0}/|\rho_0|$.  In this second part a nontrivial construction of a low norm sine polynomialis coupled by the application of the wonderful recent prime randomapproximation result of Broucke and Vindas, who sharpened the breakthroughprobabilistic construction due to Diamond, Montgomery and Vorhauer.

Oscillation of the remainder term in the prime number theorem of Beurling, "caused by a given zeta-zero"