Abstract
Let V(T) denote the number of sign changes in \(\psi (x) - x\) for \(x\in [1, T]\). We show that \(\liminf _{T\rightarrow \infty } V(T)/\log T\ge \gamma _{1}/\pi + 1.867\cdot 10^{-30}\), where \(\gamma _{1} = 14.13\ldots \) denotes the ordinate of the lowest-lying non-trivial zero of the Riemann zeta-function. This improves on a long-standing result by Kaczorowski.
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Acknowledgements
We should like to acknowledge Jerzy Kaczorowski for his valuable comments, the anonymous referee for making us aware of the result in [11], and Jonathan Bober, Florian Breuer, Ben Green, David Harvey, and Jesse Thorner for spotting a howler in the abstract.
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T. Morrill, D. Platt, and T. Trudgian: Supported by Australian Research Council Discovery Project DP160100932. T. Trudgian: Supported by Australian Research Council Future Fellowship FT160100094.
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Morrill, T., Platt, D. & Trudgian, T. Sign changes in the prime number theorem. Ramanujan J 57, 165–173 (2022). https://doi.org/10.1007/s11139-021-00398-8
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DOI: https://doi.org/10.1007/s11139-021-00398-8