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

Szilárd Gy. Révész

Published 2022-02-03Version 1

Continuing previous study of the Beurling zeta function, here we prove two results, generalizing long existing knowledge regarding the classical case of the Riemann zeta function and some of its generalizations. First, we address the question of Littlewood, who asked for explicit oscillation 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 system generated by the primes $P$, the corresponding error term $\Delta_P(x):=\psi_{P}(x)-x$, where $\psi_{P}(x)$ is the von Mangoldt summatory function 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 in the study. Finally, we prove as the next main result of the paper the following: given $\varepsilon>0$, there exists a Beurling number system with primes $P$, 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 polynomial is coupled by the application of the wonderful recent prime random approximation result of Broucke and Vindas, who sharpened the breakthrough probabilistic construction due to Diamond, Montgomery and Vorhauer.