The optimal Malliavin-type remainder for Beurling generalized integers
- Author
- Frederik Broucke (UGent) , Gregory Debruyne (UGent) and Jasson Vindas Diaz (UGent)
- Organization
- Project
- Abstract
- We establish the optimal order of Malliavin-type remainders in the asymptotic density approximation formula for Beurling generalized integers. Given α∈(0,1] and c>0 (with c≤1 if α=1 ), a generalized number system is constructed with Riemann prime counting function Π(x)=Li(x)+O(xexp(−clogαx)+log2x), and whose integer counting function satisfies the extremal oscillation estimate N(x)=ρx+Ω±(xexp(−c′(logxlog2x)αα+1) for any c′>(c(α+1))1α+1 , where ρ>0 is its asymptotic density. In particular, this improves and extends upon the earlier work [Adv. Math. 370 (2020), Article 107240].
- Keywords
- Analytic number theory, Prime number theorem, Saddle-point method, Random prime approximation, Malliavin-type error terms, Generalized integers with large oscillation, Generalized primes, PRIMES
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Citation
Please use this url to cite or link to this publication: http://hdl.handle.net/1854/LU-01GQ75S357PBJYTM5Q4FPWP5X3
- MLA
- Broucke, Frederik, et al. “The Optimal Malliavin-Type Remainder for Beurling Generalized Integers.” JOURNAL OF THE INSTITUTE OF MATHEMATICS OF JUSSIEU, vol. 23, no. 1, 2024, pp. 249–78, doi:10.1017/s147474802200038x.
- APA
- Broucke, F., Debruyne, G., & Vindas Diaz, J. (2024). The optimal Malliavin-type remainder for Beurling generalized integers. JOURNAL OF THE INSTITUTE OF MATHEMATICS OF JUSSIEU, 23(1), 249–278. https://doi.org/10.1017/s147474802200038x
- Chicago author-date
- Broucke, Frederik, Gregory Debruyne, and Jasson Vindas Diaz. 2024. “The Optimal Malliavin-Type Remainder for Beurling Generalized Integers.” JOURNAL OF THE INSTITUTE OF MATHEMATICS OF JUSSIEU 23 (1): 249–78. https://doi.org/10.1017/s147474802200038x.
- Chicago author-date (all authors)
- Broucke, Frederik, Gregory Debruyne, and Jasson Vindas Diaz. 2024. “The Optimal Malliavin-Type Remainder for Beurling Generalized Integers.” JOURNAL OF THE INSTITUTE OF MATHEMATICS OF JUSSIEU 23 (1): 249–278. doi:10.1017/s147474802200038x.
- Vancouver
- 1.Broucke F, Debruyne G, Vindas Diaz J. The optimal Malliavin-type remainder for Beurling generalized integers. JOURNAL OF THE INSTITUTE OF MATHEMATICS OF JUSSIEU. 2024;23(1):249–78.
- IEEE
- [1]F. Broucke, G. Debruyne, and J. Vindas Diaz, “The optimal Malliavin-type remainder for Beurling generalized integers,” JOURNAL OF THE INSTITUTE OF MATHEMATICS OF JUSSIEU, vol. 23, no. 1, pp. 249–278, 2024.
@article{01GQ75S357PBJYTM5Q4FPWP5X3,
abstract = {{We establish the optimal order of Malliavin-type remainders in the asymptotic density approximation formula for Beurling generalized integers. Given α∈(0,1] and c>0 (with c≤1 if α=1 ), a generalized number system is constructed with Riemann prime counting function Π(x)=Li(x)+O(xexp(−clogαx)+log2x), and whose integer counting function satisfies the extremal oscillation estimate N(x)=ρx+Ω±(xexp(−c′(logxlog2x)αα+1) for any c′>(c(α+1))1α+1 , where ρ>0
is its asymptotic density. In particular, this improves and extends upon the earlier work [Adv. Math. 370 (2020), Article 107240].}},
author = {{Broucke, Frederik and Debruyne, Gregory and Vindas Diaz, Jasson}},
issn = {{1474-7480}},
journal = {{JOURNAL OF THE INSTITUTE OF MATHEMATICS OF JUSSIEU}},
keywords = {{Analytic number theory,Prime number theorem,Saddle-point method,Random prime approximation,Malliavin-type error terms,Generalized integers with large oscillation,Generalized primes,PRIMES}},
language = {{eng}},
number = {{1}},
pages = {{249--278}},
title = {{The optimal Malliavin-type remainder for Beurling generalized integers}},
url = {{http://doi.org/10.1017/s147474802200038x}},
volume = {{23}},
year = {{2024}},
}
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