Sign changes in the prime number theorem

Abstract

Let $V(T)$ denote the number of sign changes in $\ddot{I}ˆ(x)-x$ for $x\in [1,T]$. We show that ${\mathrm{liminf}}_{T\to \infty }V(T)/\log T\ge {\hat{I}}_1^3/\ddot{I}€+1.867\cdot 10^{-30}$, where ${\hat{I}}_1^3=14.13\dots$ denotes the ordinate of the lowest-lying non-trivial zero of the Riemann zeta-function. This improves on a long-standing result by Kaczorowski.