Large deviations of the argument of the Riemann zeta function

Abstract

Let $S(t)=\frac{1}{\pi }\Im \log \zeta \left(\frac{1}{2}+it\right)$. We prove an unconditional lower bound on the measure of the sets $\{t\in [T,2T]:S(t)\ge V\}$ for $\sqrt{\log \log T}\le V\ll {\left(\frac{\log T}{\log \log T}\right)}^{1/3}$. For $V\le (\log T{)}^{1/3-\epsilon }$ our bound has a Gaussian shape with variance proportional to $\log \log T$. At the endpoint, $V\asymp {\left(\frac{\log T}{\log \log T}\right)}^{1/3}$, our result implies the best known $\Omega$-theorem for $S(t)$ which is due to Tsang. We also explain how the method breaks down for $V\gg {\left(\frac{\log T}{\log \log T}\right)}^{1/3}$ given our current knowledge about the zeros of the zeta function. Conditionally on the Riemann hypothesis we extend our results to the range $\sqrt{\log \log T}\le V\ll {\left(\frac{\log T}{\log \log T}\right)}^{1/2}$.

21 pages. The results in this version are stronger than in v1. To appear in Mathematika