Sign changes in the prime number theorem (Q2070385)

Let \(V(T)\) denote the number of sign changes in \(\psi(x)-x\) for \(x\in[1,T]\), where \(\psi(x)=\sum_{p^m\leq x}\log p\). It is known that \[ \liminf_{T\to\infty}\frac{V(T)}{\log T}\geq\frac{\gamma^\ast}{\pi}, \] where \(\gamma^\ast\) defined as follows. Let \(\Theta\) denote the supremum over \(\Re(\rho)\) where \(\rho\) ranges over all zeroes of the Riemann zeta function \(\zeta(s)\). If there are any zeroes \(\rho=\beta +i\gamma\) with \(\beta=\Theta\) we shall define \(\gamma^\ast\) as the least positive \(\gamma\); otherwise \(\gamma^\ast=\infty\). Thus, if the Riemann hypothesis is true, then \(\gamma^\ast=\gamma_1\), where \(\gamma_1\) is the imaginary part of the lowest-lying non-trivial zero of \(\zeta(s)\). If the Riemann hypothesis is false then, by a recent to appear work of the second and third authors, \(\gamma^\ast\geq 3\,000\,175\,332\,800>3\times 10^{12}\). \textit{J. Kaczorowski} [Acta Arith. 59, No. 1, 37--58 (1991; Zbl 0736.11048)] proved unconditionally that \[ \liminf_{T\to\infty}\frac{V(T)}{\log T}\geq\frac{\gamma_1}{\pi}+10^{-250}. \] In this paper, the authors follow Kaczorowski's method, making some theoretical and computational improvements, and prove the above with \(1.867\times 10^{-30}\) replacing \(10^{-250}\).