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arXiv · 2109.08509

The optimal Malliavin-type remainder for Beurling generalized integers

Frederik Broucke·Gregory Debruyne·Jasson Vindas
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Abstract

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

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Frederik Broucke, Gregory Debruyne, Jasson Vindas. 2022-08-05. The optimal Malliavin-type remainder for Beurling generalized integers. https://doi.org/10.1017/s147474802200038x

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