THE OPTIMAL MALLIAVIN-TYPE REMAINDER FOR BEURLING GENERALIZED INTEGERS

Abstract We establish the optimal order of Malliavin-type remainders in asymptotic density approximation formula for Beurling generalized integers. Given $\alpha \in (0,1]$ and $c>0$ (with $c\leq 1$ if =1$ ), a number system is constructed with Riemann prime counting function $ \Pi (x)= \operatorname {\mathrm {Li}}(x)+ O(x\exp (-c \log ^{\alpha } x ) +\log _{2}x), whose integer satisfies extremal oscillation estimate $N(x)=\rho + \Omega _{\pm }(x\exp (- c'(\log x\log _{2} x)^{\frac {\alpha }{\alpha +1}})$ any $c'>(c(\alpha +1))^{\frac {1}{\alpha +1}}$ , where $\rho>0$ its density. In particular, this improves extends upon earlier work [Adv. Math. 370 (2020), Article 107240].