Average non-vanishing of Dirichlet $L$-functions at the central point
Fuente:
arXiv
Guardado en:
| Autor principal: | |
|---|---|
| Formato: | Preprint |
| Publicado: |
2018
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866916396777078784 |
|---|---|
| author | Pratt, Kyle |
| author_facet | Pratt, Kyle |
| contents | The Generalized Riemann Hypothesis implies that at least 50% of the central values $L \left( \frac{1}{2},χ\right)$ are non-vanishing as $χ$ ranges over primitive characters modulo $q$. We show that one may unconditionally go beyond GRH, in the sense that if one averages over primitive characters modulo $q$ and averages $q$ over an interval, then at least 50.073% of the central values are non-vanishing. The proof utilizes the mollification method with a three-piece mollifier, and relies on estimates for sums of Kloosterman sums due to Deshouillers and Iwaniec.
Note: The author has been made aware of an error in this work. It seems the error can be fixed, by using a different argument, and the author will present a correction in due course. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1804_01445 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Average non-vanishing of Dirichlet $L$-functions at the central point Pratt, Kyle Number Theory The Generalized Riemann Hypothesis implies that at least 50% of the central values $L \left( \frac{1}{2},χ\right)$ are non-vanishing as $χ$ ranges over primitive characters modulo $q$. We show that one may unconditionally go beyond GRH, in the sense that if one averages over primitive characters modulo $q$ and averages $q$ over an interval, then at least 50.073% of the central values are non-vanishing. The proof utilizes the mollification method with a three-piece mollifier, and relies on estimates for sums of Kloosterman sums due to Deshouillers and Iwaniec. Note: The author has been made aware of an error in this work. It seems the error can be fixed, by using a different argument, and the author will present a correction in due course. |
| title | Average non-vanishing of Dirichlet $L$-functions at the central point |
| topic | Number Theory |
| url | https://arxiv.org/abs/1804.01445 |