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THE OPTIMAL MALLIAVIN-TYPE REMAINDER FOR BEURLING GENERALIZED INTEGERS

Published online by Cambridge University Press:  09 August 2022

Frederik Broucke
Affiliation:
Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Krijgslaan 281, 9000Gent, Belgium (fabrouck.broucke@ugent.be, gregory.debruyne@ugent.be)
Gregory Debruyne
Affiliation:
Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Krijgslaan 281, 9000Gent, Belgium (fabrouck.broucke@ugent.be, gregory.debruyne@ugent.be)
Jasson Vindas*
Affiliation:
Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Krijgslaan 281, 9000Gent, Belgium (fabrouck.broucke@ugent.be, gregory.debruyne@ugent.be)

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 {\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].

Type
Research Article
Copyright
© The Author(s), 2022. Published by Cambridge University Press

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Footnotes

F. Broucke was supported by the Ghent University BOF-grant 01J04017. G. Debruyne acknowledges support by Postdoctoral Research Fellowships of the Research Foundation–Flanders (grant number 12X9719N) and the Belgian American Educational Foundation. The latter one allowed him to do part of this research at the University of Illinois at Urbana-Champaign. J. Vindas was partly supported by Ghent University through the BOF-grant 01J04017 and by the Research Foundation–Flanders through the FWO-grant 1510119N.

References

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THE OPTIMAL MALLIAVIN-TYPE REMAINDER FOR BEURLING GENERALIZED INTEGERS
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