On the logarithm of the Riemann zeta-function near the nontrivial zeros

Çi̇çek, Fatma

doi:10.1090/tran/8426

On the logarithm of the Riemann zeta-function near the nontrivial zeros
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Trans. Amer. Math. Soc. 374 (2021), 5995-6037 Request permission

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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