Large Oscillations of the Argument of the Riemann Zeta-function

Abstract

Let $S(t)$ denote the argument of the Riemann zeta-function, defined as $$S(t)=\frac{1}{\pi }\Im \log \zeta (1/2+it).$$ Assuming the Riemann hypothesis, we prove that $$S(t)={\Omega }_{\pm }\left(\frac{\log t\log \log \log t}{\log \log t}\right).$$ This improves the classical omega results of Montgomery and matches with the $\Omega$-result obtained by Bondarenko and Seip.

9 pages. We corrected a mistake in the previous version and added Andrés Chirre as a coauthor. To appear in Bull. Lond. Math. Soc