论文标题
Selberg类的$ L $ functions的有条件估算
Conditional estimates for $L$-functions in the Selberg class
论文作者
论文摘要
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.我们还提供了有关质量数字上$ \ cl（s）$的Dirichlet系数分布的其他假设的估算。此外，通过假设$ \ cl（s）$的多项式欧拉产品表示形式，我们以$ | | 3/4-σ| \ leq 1/4-1/4-1/\ log {\ log {\ lod {\ sq | t |^|^|^{\ sdeg}} $， 1/\ log {\ log {\ left（\ sq | t |^{\ sdeg} \ right）}}}}} $和$σ= 1 $，并通过假设强$λ$ -conjecture也完全明确估计。
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.
