Extreme values of the argument of the Riemann zeta function

Alexander Dobner

Published 2021-01-05Version 1

Let $S(t) = \frac{1}{\pi}\operatorname{Im}\log\zeta\left(\frac{1}{2}+it\right)$. Using Soundararajan's resonance method we prove an unconditional lower bound on the size of the tails of the distribution of $S(t)$. In particular we reproduce the best unconditional $\Omega$ result for $S(t)$ which is due to Tsang, \[ S(t) = \Omega_\pm\left(\left(\frac{\log t}{\log \log t}\right)^{1/3}\right), \] and get a bound on how often large values of $S(t)$ occur. We also give a probabilistic argument for why this $\Omega$ result may be the best possible given our current knowledge of the zeros of the zeta function.