Zeros near $s=1$ and the constant term of $L'/L$ for $L$-functions in the Selberg class
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2020
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| _version_ | 1866918361705742336 |
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| author | Táfula, Christian |
| author_facet | Táfula, Christian |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2001_02405 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Zeros near $s=1$ and the constant term of $L'/L$ for $L$-functions in the Selberg class Táfula, Christian Number Theory 11M20, 11M41 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. |
| title | Zeros near $s=1$ and the constant term of $L'/L$ for $L$-functions in the Selberg class |
| topic | Number Theory 11M20, 11M41 |
| url | https://arxiv.org/abs/2001.02405 |