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

TL;DR

This paper improves the lower bound on the number of sign changes of the error term in the Prime Number Theorem by leveraging a new density estimate for zeros of the Riemann zeta-function, significantly advancing understanding of prime distribution fluctuations.

Contribution

It introduces a novel density estimate for zeros of the $k$-function, leading to a sharper lower bound on sign changes of $ ext{psi}(x)-x$ for large T.

Findings

Lower bound for sign changes: rac{ ext{gamma}_0}{ extpi} + rac{1}{60}

Density estimate for zeros improved by a factor of over 4×10^{21}

Enhanced understanding of fluctuations in prime counting error term

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.