Large deviations of the argument of the Riemann zeta function

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 .