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arXiv · 1910.14203

Sign changes in the prime number theorem

Thomas Morrill·Dave Platt·Tim Trudgian
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Abstract

Let $V(T)$ denote the number of sign changes in $\psi(x) - x$ for $x\in[1, T]$. We show that $\liminf_{\;T\rightarrow\infty} V(T)/\log T \geq \gamma_{1}/\pi + 1.867\cdot 10^{-30}$, 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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Thomas Morrill, Dave Platt, Tim Trudgian. 2019-10-31. Sign changes in the prime number theorem. https://arxiv.org/abs/1910.14203

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