论文标题
Riemann Zeta函数的无零区域和加权Dirichlet空间中的近似
Zero-free regions of the Riemann zeta function and approximation in weighted Dirichlet spaces
论文作者
论文摘要
我们研究了Riemann Zeta函数的无零区域$ζ$与加权dirichlet空间中的近似问题有关$ d _ { - 2} $，这与Báez-Duarte的工作以来相当于Riemann假设。的确，当$α\ in（-3，-2）$给出条件时，我们证明了标准加权dirichlet $d_α$的类似近似问题，以使条件使半Plane$ \ {s \ in \ Mathbb {c}：\ re（s）> - \ frac> - \ frac> - \ frac {α+1} $ quence $ for（此外，我们将这些结果扩展到了一个分析功能的一大批加权空间$ \ ell^p_α$。作为一个特定的实例，在极限情况下，$ p = 1 $和$α= -2 $，我们提供了质数定理的新的等价公式。
We study zero-free regions of the Riemann zeta function $ζ$ related to an approximation problem in the weighted Dirichlet space $D_{-2}$ which is known to be equivalent to the Riemann Hypothesis since the work of Báez-Duarte. We prove, indeed, that analogous approximation problems for the standard weighted Dirichlet spaces $D_α$ when $α\in (-3,-2)$ give conditions so that the half-plane $\{s \in \mathbb{C}: \Re (s) > -\frac{α+1}{2}\}$ is also zero-free for $ζ$. Moreover, we extend such results to a large family of weighted spaces of analytic functions $\ell^p_α$. As a particular instance, in the limit case $p=1$ and $α=-2$, we provide a new equivalent formulation of the Prime Number Theorem.
