Guardat en:
| Autor principal: | |
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| Format: | Preprint |
| Publicat: |
2020
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| Matèries: | |
| Accés en línia: | https://arxiv.org/abs/2001.02405 |
| Etiquetes: |
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Taula de continguts:
- Let $\mathcal{L}(s) = \sum_{n=1}^{\infty} a_n n^{-s}$ be an $L$-function in the Selberg class, and $q_{\mathcal{L}}$ its conductor. Let $\ell_0(\mathcal{L})$ be the constant term of the Laurent expansion of $\mathcal{L}'/\mathcal{L}$ at $s=1$. We show that for certain families $\mathcal{F}$ of $L$-functions in the Selberg class with polynomial Euler product: $\bullet$ If $\mathcal{L}\in\mathcal{F}$ has no zeros $β+ iγ$ with $β> 1 - δ(\log q_{\mathcal{L}})^{-1}$, $|γ| < (\log q_{\mathcal{L}})^{-1/2}$ for some absolute $δ>0$, then $\Re(\ell_0(\mathcal{L})) \ll_{\mathcal{F}} \log q_{\mathcal{L}}$; $\bullet$ If $\Re(\ell_0(\mathcal{L})) \ll \log q_{\mathcal{L}}$ for all $\mathcal{L}\in \mathcal{F}$, then there is some absolute $δ> 0$ such that $\mathcal{L}$ has no zeros $β+ iγ$ with $β> 1 - δ(\log q_{\mathcal{L}})^{-1}$, $|γ| < (1-β)^{1/2}(\log q_{\mathcal{L}})^{-1/2}$. This generalizes, for instance, the case of families of Dedekind zeta functions of number fields with bounded degree.