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

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 $$\underset{T\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}inf}}\frac{V(T)}{\log T}\ge \frac{{\gamma }_0}{\pi }+\frac{1}{60}$$ 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.