On the number of sign changes of the function \(S(t)\) on a short interval (Q859104)

For a real number \(t\) different from the imaginary part of a zero of the Riemann zeta function \(\zeta(s)\), \[ S(t)= \tfrac1\pi\arg \zeta\bigl(\tfrac12+it\bigr), \] where \(\arg \zeta(\frac12+it)\) is obtained by a continuous continuation along the polygonal line joining the point \(s=2\), \(s=2+it\), and \(s=\frac12+it\) of the branch of the argument of \(\zeta(s)\) for which \(\arg\zeta(2)=0\). Let \(N_1(T)\) be the number of sign changes of \(S(t)\) on the interval \((0;T]\). Using the results of \textit{A. Selberg} [Arch. Math. Naturvid. 48, No. 5, 89--155 (1946; Zbl 0061.08402)], \textit{A. Ghosh} [in: Recent progress in analytic number theory, Symp. Durham 1979, Vol. 1, 25--46 (1981; Zbl 0457.10018)] and \textit{A. A. Karatsuba} [Izv. Math. 60, No. 5, 901--931 (1996); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 60, No. 5, 27--56 (1996; Zbl 0898.11032)], this author proved the following assertion. Theorem. Suppose that \(0<\alpha<0.001\) is an arbitrary fixed number and \(H=T^{\frac{27}{82}+\alpha}\). Then there exist positive constants \(T_1= T_1(\alpha)\) and \(c_1=c_1(\alpha)\) such that, at \(T\geq T_1\), we have \[ N_1(T+H)-N_1(T)\geq H\ln T\exp \biggl(- \frac{c\ln\ln T}{\sqrt{\ln\ln\ln T}} \biggr). \] Corollary. There exist absolute positive constants \(T_0\) and \(c_0\) such that, at \(T\geq T_0\), we have \[ N_1(T)\geq T\ln T\exp\biggl(- \frac{c_0\ln\ln T}{\sqrt{\ln\ln\ln T}}\biggr). \]