论文标题
临界线附近Riemann Zeta功能的对数衍生物的平均值
Mean values of the logarithmic derivative of the Riemann zeta-function near the critical line
论文作者
论文摘要
假设Riemann的假设和关于Zeta Zeros之间较小差距的假设，我们证明了Bailey，Bettin，Bebtin，Blower，Conrey，Prokhorov，Prokhorov，Rubinstein和Snaith的猜想，这指出了任何积极的integer $ k $ $ k $和实际数量$ k $和实数$ a> 0 $ a> $ a> n $ a> weben \ feben \ dim__ and and and {allim_}+lim_} \ to \ to \ infty} \ frac {（2a）^{2k-1}}} {t（\ log t）^{2K}} \ int_ {t}^{2t}^{2t} \ left | \ frac {prac {prac {prac {ζ'} em'} em \ frac \ left（ t}+it \ right）\ right |^{2k} dt = \ binom {2K-2} {k-1}。 \ end {align*}当$ k = 1 $时，这本质上是戈德斯顿，dogek和蒙哥马利的结果。
Assume the Riemann Hypothesis and a hypothesis on small gaps between zeta zeros, we prove a conjecture of Bailey, Bettin, Blower, Conrey, Prokhorov, Rubinstein and Snaith, which states that for any positive integer $K$ and real number $a>0$, \begin{align*} \lim_{a \to 0^+}\lim_{T \to \infty} \frac{(2a)^{2K-1}}{T (\log T)^{2K}} \int_{T}^{2T} \left|\frac{ζ'}ζ\left(\frac{1}{2}+\frac{a}{\log T}+it\right)\right|^{2K} dt = \binom{2K-2}{K-1}. \end{align*} When $K=1$, this was essentially a result of Goldston, Gonek and Montgomery.
