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

Abstract

Continuing previous studies 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 $\zeta _{{\mathcal {P}}}$ for a given number system generated by the primes ${\mathcal {P}}$, the corresponding error term $\Delta (x):=\psi _{{\mathcal {P}}}(x)-x$, where $\psi _{{\mathcal {P}}}(x)$ is the von Mangoldt summatory function shows oscillation in any large enough interval, as large as $\frac {\pi /2-\varepsilon }{|\rho _0|}x^{\Re \rho _0}$. 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 ${\mathcal {P}}$, such that $|\Delta (x)| \le \frac {\pi /2+\varepsilon }{|\rho _0|}x^{\Re \rho _0}$. In this 2nd 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.