The optimal malliavin-type remainder for beurling generalized integers

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