论文标题
在Riemann Zeta功能的对数附近，非平凡零
On the Logarithm of the Riemann Zeta-function Near the Nontrivial Zeros
论文作者
论文摘要
假设Riemann假设和蒙哥马利的一对相关构宗，我们研究了序列的分布$（\ log | pem |ζ（ρ+z）|）$和$（\argζ（ρ+z））。$ $ρ= \ frac12+iγ$在$ trivial zeros of the trivial zeros $ $ 0 <大量实际数字，$ z = u+iv $是一个非零复合数的模量$ \ ll 1/\ logt。$我们的方法通过研究这些序列的整体矩来进行。如果我们让$ z $倾向于$ 0 $，并且进一步假设所有零$ρ$都很简单，我们可以用对零上的较弱的间距假设替换对零的间距假说，并推断该序列$（\ log log（|ζ^\ prime（ρ）|/\ log t t）$ y Moguss $ $ 0 $ 0 $ 0 $ 0 $ $ 0 $ $ $ 0。这提供了赫贾尔旧结果的替代证明，并通过提供分配的融合率来改善它。
Assuming the Riemann hypothesis and Montgomery's Pair Correlation Conjecture, we investigate the distribution of the sequences $(\log|ζ(ρ+z)|)$ and $(\argζ(ρ+z)).$ Here $ρ=\frac12+iγ$ runs over the nontrivial zeros of the zeta-function, $0<γ\leq T,$ $T$ is a large real number, and $z=u+iv$ is a nonzero complex number of modulus $\ll 1/\log T.$ Our approach proceeds via a study of the integral moments of these sequences. If we let $z$ tend to $0$ and further assume that all the zeros $ρ$ are simple, we can replace the pair correlation conjecture with a weaker spacing hypothesis on the zeros and deduce that the sequence $(\log (|ζ^\prime(ρ)|/\log T))$ has an approximate Gaussian distribution with mean $0$ and variance $\frac12\log\log T.$ This gives an alternative proof of an old result of Hejhal and improves it by providing a rate of convergence to the distribution.
