Large values of quadratic Dirichlet $L$-functions
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866914826827071488 |
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| author | Darbar, Pranendu Maiti, Gopal |
| author_facet | Darbar, Pranendu Maiti, Gopal |
| contents | Assuming the Generalized Riemann Hypothesis (GRH), we utilize the long resonator method to derive $Ω$-results for the family of quadratic Dirichlet $L$-functions $L(σ, χ_d)$, where $d$ runs over all fundamental discriminants with $|d| \leq X$ and $σ\in [1/2, 1]$ is fixed. This study advances understanding of the maximum size of $L(σ, χ_d)$ within the segment $σ\in [1/2, 1]$. In particular, we improve upon Soundararajan's results at the central point and provide a lower bound on the proportion of fundamental discriminants, uniformly within an expected order of magnitude, up to optimal values of the constant for a fixed $σ\in (1/2, 1]$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_01519 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Large values of quadratic Dirichlet $L$-functions Darbar, Pranendu Maiti, Gopal Number Theory 11A05, 11L40, 11M06, 11M20, 11N37 Assuming the Generalized Riemann Hypothesis (GRH), we utilize the long resonator method to derive $Ω$-results for the family of quadratic Dirichlet $L$-functions $L(σ, χ_d)$, where $d$ runs over all fundamental discriminants with $|d| \leq X$ and $σ\in [1/2, 1]$ is fixed. This study advances understanding of the maximum size of $L(σ, χ_d)$ within the segment $σ\in [1/2, 1]$. In particular, we improve upon Soundararajan's results at the central point and provide a lower bound on the proportion of fundamental discriminants, uniformly within an expected order of magnitude, up to optimal values of the constant for a fixed $σ\in (1/2, 1]$. |
| title | Large values of quadratic Dirichlet $L$-functions |
| topic | Number Theory 11A05, 11L40, 11M06, 11M20, 11N37 |
| url | https://arxiv.org/abs/2406.01519 |