The optimal Malliavin-type remainder for Beurling generalized integers
Fuente:
arXiv
Guardado en:
| Autores principales: | , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2021
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866910386971738112 |
|---|---|
| author | Broucke, Frederik Debruyne, Gregory Vindas, Jasson |
| author_facet | Broucke, Frederik Debruyne, Gregory Vindas, Jasson |
| contents | 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]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2109_08509 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | The optimal Malliavin-type remainder for Beurling generalized integers Broucke, Frederik Debruyne, Gregory Vindas, Jasson Number Theory Complex Variables 11M41, 11N80, 11N05 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]. |
| title | The optimal Malliavin-type remainder for Beurling generalized integers |
| topic | Number Theory Complex Variables 11M41, 11N80, 11N05 |
| url | https://arxiv.org/abs/2109.08509 |