论文标题
接近$ s = 1 $的零和$ l'/l $ for $ l $ for $ l $ fors selberg class
Zeros near $s=1$ and the constant term of $L'/L$ for $L$-functions in the Selberg class
论文作者
论文摘要
令$ \ mathcal {l}（s）= \ sum_ {n = 1}^{\ infty} a_n n^{ - s} $是Selberg类中的$ L $ - 功能，而$ Q _ {\ Mathcal {L}} $。令$ \ ell_0（\ Mathcal {l}）$为$ \ Mathcal {l}'/\ Mathcal {l} $ at $ s = 1 $的laurent扩展的常数术语。我们表明，对于某些家庭，$ \ Mathcal {f} $ $ l $ functions在Selberg类中使用多项式Euler产品： $ \ bullet $如果$ \ mathcal {l} \ in \ mathcal {f} $没有零$β+Iγ$，$β+Iγ$，$β> 1-δ（\ log q _ {\ nathcal {l}}}） <（\ log q _ {\ Mathcal {l}}）^{ - 1/2} $对于某些绝对$δ> 0 $，然后$ \ re（\ ell_0（\ ell_0（\ Mathcal {l}）） $ \ bullet $如果$ \ re（\ ell_0（\ Mathcal {l}）））\ ll \ log q _ {\ Mathcal {l}} $ for las $ \ Mathcal {l} \ in \ Mathcal in \ Mathcal {f} Iγ$带$β> 1-δ（\ log q _ {\ Mathcal {l}}}）^{ - 1} $，$ |γ| <（1-β）^{1/2}（\ log q _ {\ Mathcal {l}}}）^{ - 1/2} $。 例如，这将概括为具有界限的数字字段的Dedekind Zeta函数家族的情况。
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.
