The optimal Malliavin-type remainder for Beurling generalized integers

TL;DR

This paper constructs a generalized number system with optimal Malliavin-type remainders, improving the understanding of asymptotic density approximation for Beurling generalized integers and extending previous results.

Contribution

It establishes the optimal order of Malliavin-type remainders for Beurling generalized integers, providing explicit constructions and extremal oscillation estimates.

Findings

Constructed a generalized number system with specified Riemann prime counting function.

Proved the extremal oscillation estimate for the integer counting function.

Extended previous work by improving the order of remainders and oscillation bounds.

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 $c\leq 1$ if $\alpha=1$), a generalized number system is constructed with Riemann prime counting function $ \Pi(x)= \operatorname*{Li}(x)+ O(x\exp (-c \log^{\alpha} x ) +\log_{2}x), $ and whose integer counting function satisfies the extremal oscillation estimate $N(x)=\rho x + \Omega_{\pm}(x\exp(- c'(\log x\log_{2} x)^{\frac{\alpha}{\alpha+1}})$ for any $c'>(c(\alpha+1))^{\frac{1}{\alpha+1}}$, where $\rho>0$ is its asymptotic density. In particular, this improves and extends upon the earlier work [Adv. Math. 370 (2020), Article 107240].