Conditional estimates for $L$-functions in the Selberg class
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866913467914518528 |
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| author | Palojärvi, Neea Simonič, Aleksander |
| author_facet | Palojärvi, Neea Simonič, Aleksander |
| contents | Assuming the Generalized Riemann Hypothesis, we provide uniform upper bounds with explicit main terms for moduli of $\left(\cL'/\cL\right)(s)$ and $\log{\cL(s)}$ for $1/2+δ\leqσ<1$, fixed $δ\in(0,1/2)$ and for functions in the Selberg class except for the identity function. We also provide estimates under additional assumptions on the distribution of Dirichlet coefficients of $\cL(s)$ on prime numbers. Moreover, by assuming a polynomial Euler product representation for $\cL(s)$, we establish uniform bounds for $|3/4-σ|\leq 1/4-1/\log{\log{\left(\sq|t|^{\sdeg}\right)}}$, $|1-σ|\leq 1/\log{\log{\left(\sq|t|^{\sdeg}\right)}}$ and $σ=1$, and completely explicit estimates by assuming also the strong $λ$-conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_01121 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Conditional estimates for $L$-functions in the Selberg class Palojärvi, Neea Simonič, Aleksander Number Theory 11M06, 11M26, 41A30 Assuming the Generalized Riemann Hypothesis, we provide uniform upper bounds with explicit main terms for moduli of $\left(\cL'/\cL\right)(s)$ and $\log{\cL(s)}$ for $1/2+δ\leqσ<1$, fixed $δ\in(0,1/2)$ and for functions in the Selberg class except for the identity function. We also provide estimates under additional assumptions on the distribution of Dirichlet coefficients of $\cL(s)$ on prime numbers. Moreover, by assuming a polynomial Euler product representation for $\cL(s)$, we establish uniform bounds for $|3/4-σ|\leq 1/4-1/\log{\log{\left(\sq|t|^{\sdeg}\right)}}$, $|1-σ|\leq 1/\log{\log{\left(\sq|t|^{\sdeg}\right)}}$ and $σ=1$, and completely explicit estimates by assuming also the strong $λ$-conjecture. |
| title | Conditional estimates for $L$-functions in the Selberg class |
| topic | Number Theory 11M06, 11M26, 41A30 |
| url | https://arxiv.org/abs/2211.01121 |