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THE OPTIMAL MALLIAVIN-TYPE REMAINDER FOR BEURLING GENERALIZED INTEGERS
- Part of
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 23 / Issue 1 / January 2024
- Published online by Cambridge University Press:
- 09 August 2022, pp. 249-278
- Print publication:
- January 2024
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- Article
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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 {\mathrm {Li}}(x)+ O(x\exp (-c \log ^{\alpha } x ) +\log _{2}x), $ and whose integer counting function satisfies the extremal oscillation estimate 
$N(x)=\rho x + \Omega _{\pm }(x\exp (- c'(\log x\log _{2} x)^{\frac {\alpha }{\alpha +1}})$ 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].
