Large deviations of the argument of the Riemann zeta function

Abstract

Let $$. We prove an unconditional lower bound on the measure of the sets $$ for $$. For $$, our bound has a Gaussian shape with variance proportional to $$. At the endpoint, $$, our result implies the best known $$-theorem for $$ that is due to Tsang. We also explain how the method breaks down for $$ given our current knowledge about the zeros of the zeta function. Conditionally on the Riemann hypothesis, we extend our results to the range $$.