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arXiv · 2408.10399

On the sign changes of $\psi(x)-x$

Maciej Grześkowiak·Jerzy Kaczorowski·Łukasz Pańkowski·Maciej Radziejewski
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Abstract

We improve the lower bound for $V(T)$, the number of sign changes of the error term $\psi(x)-x$ in the Prime Number Theorem in the interval $[1,T]$ for large $T$. We show that \[ \liminf_{T\to\infty}\frac{V(T)}{\log T}\geq\frac{\gamma_{0}}{\pi}+\frac{1}{60} \] where $\gamma_{0}=14.13\ldots$ is the imaginary part of the lowest-lying non-trivial zero of the Riemann zeta-function. The result is based on a new density estimate for zeros of the associated $k$-function, over $4\cdot10^{21}$ times better than previously known estimates of this type.

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Maciej Grześkowiak, Jerzy Kaczorowski, Łukasz Pańkowski, Maciej Radziejewski. 2024-08-19. On the sign changes of $\psi(x)-x$. https://doi.org/10.1090/mcom/4053

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