The optimal Malliavin-type remainder for Beurling generalized integers

Abstract

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

27 pages

References in corpus (2)

• Beurling integers with RH and large oscillation
• Note on a conjecture of Bateman and Diamond concerning the abstract PNT with Malliavin-type remainder

Cited by in corpus (2)

• A Mean Value Theorem for general Dirichlet Series
• Some examples of well-behaved Beurling number systems