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We assume the Riemann hypothesis to improve upon the rate of convergence of \((\log \log \log T)^2/\sqrt{\log \log T}\) in Selberg’s central limit theorem for \(\log |\zeta (1/2+it)|\) given by the author in [8]. We achieve a rate of convergence of \(\sqrt{\log \log \log \log T}/\sqrt{\log \log T}\) in the Dudley distance. The proof is an adaptation of the techniques used by the author in [8], based on the work of Radziwiłł and Soundararajan in [7] and Arguin et al. in [1], combined with a lemma of Selberg [11] that provides for a mollifier close to the critical line \({{\,\textrm{Re}\,}}(s)=1/2\) under the Riemann hypothesis.

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The Rate of Convergence for Selberg’s Central Limit Theorem Under the Riemann Hypothesis

  • 会议论文

  • 发表日期：

    2025年

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  • Asher Roberts

摘要

We assume the Riemann hypothesis to improve upon the rate of convergence of \((\log \log \log T)^2/\sqrt{\log \log T}\) in Selberg’s central limit theorem for \(\log |\zeta (1/2+it)|\) given by the author in [8]. We achieve a rate of convergence of \(\sqrt{\log \log \log \log T}/\sqrt{\log \log T}\) in the Dudley distance. The proof is an adaptation of the techniques used by the author in [8], based on the work of Radziwiłł and Soundararajan in [7] and Arguin et al. in [1], combined with a lemma of Selberg [11] that provides for a mollifier close to the critical line \({{\,\textrm{Re}\,}}(s)=1/2\) under the Riemann hypothesis.