Note on a conjecture of Bateman and Diamond concerning the abstract PNT with Malliavin-type remainder

Abstract

Given $\beta \in (0,1)$, we show the existence of a Beurling generalized number system whose integer counting satisfies $N(x)=ax+O(x\exp (-c{\log }^{\beta }x))$ for some $a>0$ and $c>0$, and whose prime counting function satisfies $\pi (x)=\mathrm{Li}(x)+\Omega (x\exp (-c^{\prime }(\log x{)}^{\frac{\beta }{\beta +1}}))$ for some $c^{\prime }>0$. This is done by generalizing a construction of Diamond, Montgomery, and Vorhauer. This Beurling system serves as additional motivation for a conjecture of Bateman and Diamond from 1969, concerning the PNT with Malliavin-type remainder.

11 pages

References in corpus (1)

• Beurling integers with RH and large oscillation

Cited by in corpus (1)

• The optimal Malliavin-type remainder for Beurling generalized integers