Large deviations of the argument of the Riemann zeta function

TL;DR

This paper establishes lower bounds on the measure of large deviations of the argument of the Riemann zeta function, revealing Gaussian behavior for certain ranges and extending results under the Riemann hypothesis.

Contribution

It provides the first unconditional lower bounds for large deviations of the argument of the zeta function and extends these results conditionally on the Riemann hypothesis.

Findings

Unconditional lower bounds match Gaussian distribution for certain V.

Best known Omega-theorem for S(t) at the critical endpoint.

Method limitations explained for V beyond a certain threshold.

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 $\sqrt{\log\log T} \leq V \ll \left(\frac{\log T}{\log \log T}\right)^{1/2}$.