Extreme values of the Riemann zeta function and its argument

Abstract

We combine our version of the resonance method with certain convolution formulas for $ζ(s)$ and $\log\, ζ(s)$. This leads to a new $Ω$ result for $|ζ(1/2+it)|$: The maximum of $|ζ(1/2+it)|$ on the interval $1 \le t \le T$ is at least $\exp\left((1+o(1)) \sqrt{\log T \log\log\log T/\log\log T}\right)$. We also obtain conditional results for $S(t):=1/π$ times the argument of $ζ(1/2+it)$ and $S_1(t):=\int_0^t S(τ)dτ$. On the Riemann hypothesis, the maximum of $|S(t)|$ is at least $c \sqrt{\log T \log\log\log T/\log\log T}$ and the maximum of $S_1(t)$ is at least $c_1 \sqrt{\log T \log\log\log T/(\log\log T)^3}$ on the interval $T^β \le t \le T$ whenever $0\le β< 1$.