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

Given β ∈ (0 , 1) , we show the existence of a Beurling generalized number system whose integer counting satisfies N (x) = a x + O (x exp (- c log β x)) for some a > 0 and c > 0 , and whose prime counting function satisfies π (x) = Li (x) + Ω (x exp (- c ′ (log x) β β + 1 )) for some c ′ > 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 prime number theorem with Malliavin-type remainder. [ABSTRACT FROM AUTHOR]