Oscillation of the Remainder Term in the Prime Number Theorem of Beurling, "Caused by a Given ζ-Zero".

The article focuses on extending classical oscillation results related to the error term in the prime number theorem (PNT) to the context of Beurling generalized number systems, which generalize the natural numbers and primes. It proves that given a zero \(\rho_0\) of the Beurling zeta function \(\zeta_{\mathcal{P}}\) associated with a Beurling prime system \(\mathcal{P}\), the corresponding error term \(\Delta(x) = \psi_{\mathcal{P}}(x) - x\) exhibits oscillations of size at least \((\pi/2 - \varepsilon) x^{\Re \rho_0} / |\rho_0|\) for arbitrarily large \(x\). Moreover, the authors construct explicit Beurling number systems satisfying Axiom A for which the oscillation magnitude is bounded above by \((\pi/2 + \varepsilon) x^{\Re \rho_0} / |\rho_0|\), demonstrating the optimality of the constant \(\pi/2\). The work employs advanced analytic techniques, including a modified Cassels power-sum theorem, contour integration, and recent probabilistic prime approximation results by Broucke and Vindas, to control the zero distribution and error terms. Additionally, the paper outlines a framework for constructing Beurling prime systems with prescribed zeros and discusses forthcoming generalizations of zero-free region results and their implications for error bounds in the Beurling prime number theorem.