Density results for $r$-gaps between zeros of the Riemann zeta-function
Fuente:
arXiv
Guardado en:
| Autor principal: | |
|---|---|
| Formato: | Preprint |
| Publicado: |
2026
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866915871435259904 |
|---|---|
| author | Zhao, Tianyu |
| author_facet | Zhao, Tianyu |
| contents | Let $0<γ_1\leq γ_2\leq \ldots$ denote the positive ordinates of the non-trivial zeros of the Riemann zeta-function. A result first announced by Selberg states that there exist absolute constants $Θ, \vartheta>0$ such that for each $r\in \mathbb{N}$, \[ \limsup_{n\to \infty}\frac{γ_{n+r}-γ_n}{2πr/\log γ_n}\geq 1+\fracΘ{r^α} \qquad \text{and}\qquad \liminf_{n\to \infty}\frac{γ_{n+r}-γ_n}{2πr/\log γ_n}\leq 1-\frac{\vartheta}{r^α} \] where $α$ may be taken as $2/3$, or as $1/2$ if one assumes the Riemann hypothesis. This was recently proved by Conrey and Turnage-Butterbaugh under RH and by Inoue unconditionally. We prove that in fact a positive proportion of $r$-gaps are large (and small) to the above extent, and we provide explicit estimates for the sizes and proportions of these gaps. In the case $r=1$, this quantitatively improves an unconditional result of Simonič, Trudgian and Turnage-Butterbaugh. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_17334 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Density results for $r$-gaps between zeros of the Riemann zeta-function Zhao, Tianyu Number Theory 11M06, 11M26 Let $0<γ_1\leq γ_2\leq \ldots$ denote the positive ordinates of the non-trivial zeros of the Riemann zeta-function. A result first announced by Selberg states that there exist absolute constants $Θ, \vartheta>0$ such that for each $r\in \mathbb{N}$, \[ \limsup_{n\to \infty}\frac{γ_{n+r}-γ_n}{2πr/\log γ_n}\geq 1+\fracΘ{r^α} \qquad \text{and}\qquad \liminf_{n\to \infty}\frac{γ_{n+r}-γ_n}{2πr/\log γ_n}\leq 1-\frac{\vartheta}{r^α} \] where $α$ may be taken as $2/3$, or as $1/2$ if one assumes the Riemann hypothesis. This was recently proved by Conrey and Turnage-Butterbaugh under RH and by Inoue unconditionally. We prove that in fact a positive proportion of $r$-gaps are large (and small) to the above extent, and we provide explicit estimates for the sizes and proportions of these gaps. In the case $r=1$, this quantitatively improves an unconditional result of Simonič, Trudgian and Turnage-Butterbaugh. |
| title | Density results for $r$-gaps between zeros of the Riemann zeta-function |
| topic | Number Theory 11M06, 11M26 |
| url | https://arxiv.org/abs/2603.17334 |