Abstract:Assuming the Riemann hypothesis and Montgomery’s Pair Correlation Conjecture, we investigate the distribution of the sequences $(\log |\zeta (\rho +z)|)$ and $(\arg \zeta (\rho +z)).$ Here $\rho =\frac 12+i\gamma$ runs over the nontrivial zeros of the zeta-function, $0<\gamma \leq T,$ $T$ 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 $\rho$ are simple, we can replace the pair correlation conjecture with a weaker spacing hypothesis on the zeros and deduce that the sequence $(\log ( |\zeta ^\prime (\rho )|/\log T))$ has an approximate Gaussian distribution with mean $0$ and variance $\tfrac 12\log \log T.$ This gives an alternative proof of an old result of Hejhal and improves it by providing a rate of convergence to the distribution.

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中文翻译：

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关于非平凡零点附近黎曼 zeta 函数的对数

摘要：假设黎曼假设和蒙哥马利对相关猜想，我们研究了序列 $(\log |\zeta (\rho +z)|)$ 和 $(\arg \zeta (\rho +z)) 的分布。 $ 这里 $\rho =\frac 12+i\gamma$ 运行在 zeta 函数的非平凡零点上，$0<\gamma \leq T，$ $T$ 是一个大实数，而 $z=u+iv $ 是模数 $\ll 1/\log T.$ 的非零复数。我们的方法是通过研究这些序列的积分矩来进行的。如果我们让 $z$ 趋向于 $0$ 并进一步假设所有的零 $\rho$ 都是简单的，我们可以用一个较弱的零间距假设代替对相关猜想，并推导出序列 $(\log ( | \zeta ^\prime (\rho )|/\log T))$ 具有近似高斯分布，均值为 $0$，方差为 $\tfrac 12\log \log T。

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