Simultaneous nonvanishing of Dirichlet $L$-functions in Galois orbits
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866916834300657664 |
|---|---|
| author | Bui, Hung M. Florea, Alexandra Ngo, Hieu T. |
| author_facet | Bui, Hung M. Florea, Alexandra Ngo, Hieu T. |
| contents | Under the Generalized Riemann Hypothesis, we prove that given any two distinct imprimitive Dirichlet characters $η_1, η_2$ modulo $q=p^k$, a positive proportion of characters $χ$ modulo $q$ in a fixed Galois orbit of primitive characters satisfies the nonvanishing property that $L(1/2,χη_1) L(1/2,χη_2) \neq 0$, as $k \to \infty$ (with $p$ fixed). Previously, only a positive proportion of nonvanishing result was available in Galois orbits (as opposed to simultaneously nonvanishing), due to work of Khan, Milićević and Ngo. The main ingredients are obtaining a sharp upper bound on the mollified fourth moment over the Galois orbit using an Euler product mollifier, and obtaining a lower bound for the mollified second moment, which relies on using results from Diophantine approximation (such as the $p$-adic Roth theorem). We also unconditionally compute the second moments for $L$--functions associated to primitive Dirichlet characters in full orbits and thinner orbits. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_06609 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Simultaneous nonvanishing of Dirichlet $L$-functions in Galois orbits Bui, Hung M. Florea, Alexandra Ngo, Hieu T. Number Theory Under the Generalized Riemann Hypothesis, we prove that given any two distinct imprimitive Dirichlet characters $η_1, η_2$ modulo $q=p^k$, a positive proportion of characters $χ$ modulo $q$ in a fixed Galois orbit of primitive characters satisfies the nonvanishing property that $L(1/2,χη_1) L(1/2,χη_2) \neq 0$, as $k \to \infty$ (with $p$ fixed). Previously, only a positive proportion of nonvanishing result was available in Galois orbits (as opposed to simultaneously nonvanishing), due to work of Khan, Milićević and Ngo. The main ingredients are obtaining a sharp upper bound on the mollified fourth moment over the Galois orbit using an Euler product mollifier, and obtaining a lower bound for the mollified second moment, which relies on using results from Diophantine approximation (such as the $p$-adic Roth theorem). We also unconditionally compute the second moments for $L$--functions associated to primitive Dirichlet characters in full orbits and thinner orbits. |
| title | Simultaneous nonvanishing of Dirichlet $L$-functions in Galois orbits |
| topic | Number Theory |
| url | https://arxiv.org/abs/2507.06609 |