The Argument of the Riemann Zeta Function on the Critical Line

M. A. Korolev

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Discrete geometry and geometry of numbers, Tome 239 (2002), pp. 215-238

Résumé

The following assertion is proved: If $N_1(T)$ is the number of sign changes of the argument of the Riemann zeta function $\zeta (s)$ on the interval $0\operatorname{Im}s\le T$ of the critical line$\operatorname{Re}s=1/2$, then, for any $a$ such that $27/82$, $T\ge T_1(a)>0$, and $H=T^a$, the inequality $N_1(T+H)-N_1(T) \ge H\log T\exp \bigl (-\frac {c\log \log T}{\sqrt {\log \log \log T}}\bigr )$ holds with a constant $c=c(a)>0$.