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 $\{t\in [T,2T] \colon S(t) \geq V\}$ for $\sqrt{\log\log T} \leq V \ll \left(\frac{\log T}{\log \log T}\right)^{1/3}$. For $V \leq (\log T)^{1/3-\varepsilon}$ 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 $\sqrt{\log\log T} \leq V \ll \left(\frac{\log T}{\log \log T}\right)^{1/2}$.