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 \[ \liminf _{T\to \infty }\frac {V(T)}{\log T}\geq \frac {\gamma _{0}}{\pi }+\frac {1}{60}, \] 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 $4\cdot 10^{21}$ times better than previously known estimates of this type.