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

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 \[ \liminf_{T\to\infty}\frac{V(T)}{\log T}\geq\frac{γ_{0}}π+\frac{1}{60} \] where ${\gamma }_0=14.13\dots$ 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 $4\cdot 10^{21}$ times better than previously known estimates of this type.